GCSE Maths: Vectors: column vectors, addition and scalar multiples (Foundation)

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Count the squares from the tail of the arrow to its tip: across first, then up or down.

1. Find the move across. Right is positive and left is negative. Count from the start of the arrow to its tip, sideways only.
   3 right: top number = 3
2. Find the move up or down. Up is positive and down is negative.
   2 up: bottom number = 2
3. Write the column vector. Across on top, up underneath.
   a = 3
   2

Check. Start at the tail and add the vector: move 3 right and 2 up. You arrive at the tip. ✓

Watch out. The top number is across (x) and the bottom number is up (y). Do not swap them.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Count the squares from the tail of the arrow to its tip: across first, then up or down.

1. Find the move across. Right is positive and left is negative. Count from the start of the arrow to its tip, sideways only.
   2 left: top number = −2
2. Find the move up or down. Up is positive and down is negative.
   3 up: bottom number = 3
3. Write the column vector. Across on top, up underneath.
   b = −2
   3

Check. Start at the tail and add the vector: move 2 left and 3 up. You arrive at the tip. ✓

Watch out. The top number is across (x) and the bottom number is up (y). Do not swap them.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Count the squares from the tail of the arrow to its tip: across first, then up or down.

1. Find the move across. Right is positive and left is negative. Count from the start of the arrow to its tip, sideways only.
   no sideways move: top number = 0
2. Find the move up or down. Up is positive and down is negative.
   3 down: bottom number = −3
3. Write the column vector. Across on top, up underneath.
   c = 0
   −3

Check. Start at the tail and add the vector: move no sideways move and 3 down. You arrive at the tip. ✓

Watch out. The top number is across (x) and the bottom number is up (y). Do not swap them.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Count the squares from the tail of the arrow to its tip: across first, then up or down.

1. Find the move across. Right is positive and left is negative. Count from the start of the arrow to its tip, sideways only.
   4 right: top number = 4
2. Find the move up or down. Up is positive and down is negative.
   no up or down move: bottom number = 0
3. Write the column vector. Across on top, up underneath.
   d = 4
   0

Check. Start at the tail and add the vector: move 4 right and no up or down move. You arrive at the tip. ✓

Watch out. The top number is across (x) and the bottom number is up (y). Do not swap them.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Right and up are positive. Left and down are negative. Put the sideways move on top and the up-down move underneath.

1. Write the sideways move. Right = positive, left = negative. If there is no sideways move, write 0.
   3 right → top number 3
2. Write the up-down move. Up = positive, down = negative. If there is none, write 0.
   2 up → bottom number 2
3. Write the column vector.
   3
   2

Check. Reading it back: 3
2 means 3 right and 2 up. ✓

Watch out. Left and down need a minus sign.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Right and up are positive. Left and down are negative. Put the sideways move on top and the up-down move underneath.

1. Write the sideways move. Right = positive, left = negative. If there is no sideways move, write 0.
   4 left → top number −4
2. Write the up-down move. Up = positive, down = negative. If there is none, write 0.
   1 up → bottom number 1
3. Write the column vector.
   −4
   1

Check. Reading it back: −4
1 means 4 left and 1 up. ✓

Watch out. Left and down need a minus sign.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Right and up are positive. Left and down are negative. Put the sideways move on top and the up-down move underneath.

1. Write the sideways move. Right = positive, left = negative. If there is no sideways move, write 0.
   2 right → top number 2
2. Write the up-down move. Up = positive, down = negative. If there is none, write 0.
   5 down → bottom number −5
3. Write the column vector.
   2
   −5

Check. Reading it back: 2
−5 means 2 right and 5 down. ✓

Watch out. Left and down need a minus sign.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Right and up are positive. Left and down are negative. Put the sideways move on top and the up-down move underneath.

1. Write the sideways move. Right = positive, left = negative. If there is no sideways move, write 0.
   6 left → top number −6
2. Write the up-down move. Up = positive, down = negative. If there is none, write 0.
   no up-down move → bottom number 0
3. Write the column vector.
   −6
   0

Check. Reading it back: −6
0 means 6 left. ✓

Watch out. Left and down need a minus sign.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Subtract the coordinates of the start point from the coordinates of the end point.

1. Write the two points. A column vector from A to B says how far to move across and up to get from A to B.
   A = (1, 2) and B = (4, 6)
2. Find the across move. Subtract the x-coordinates: end minus start.
   4 − 1 = 3
3. Find the up move. Subtract the y-coordinates: end minus start.
   6 − 2 = 4
4. Write the column vector. Across on top, up underneath.
   →AB = 3
   4

Check. Start at A and add the vector: (1 + 3, 2 + 4) = (4, 6) = B. ✓

Watch out. It is end minus start. If you swap them you get the vector going the other way, which has the opposite signs.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Subtract the coordinates of the start point from the coordinates of the end point.

1. Write the two points. A column vector from A to B says how far to move across and up to get from A to B.
   A = (−2, 3) and B = (2, −1)
2. Find the across move. Subtract the x-coordinates: end minus start.
   2 − (−2) = 4
3. Find the up move. Subtract the y-coordinates: end minus start.
   −1 − 3 = −4
4. Write the column vector. Across on top, up underneath.
   →AB = 4
   −4

Check. Start at A and add the vector: (−2 + 4, 3 + −4) = (2, −1) = B. ✓

Watch out. It is end minus start. If you swap them you get the vector going the other way, which has the opposite signs.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Subtract the coordinates of the start point from the coordinates of the end point.

1. Write the two points. A column vector from A to B says how far to move across and up to get from A to B.
   A = (5, 4) and B = (1, 4)
2. Find the across move. Subtract the x-coordinates: end minus start.
   1 − 5 = −4
3. Find the up move. Subtract the y-coordinates: end minus start.
   4 − 4 = 0
4. Write the column vector. Across on top, up underneath.
   →AB = −4
   0

Check. Start at A and add the vector: (5 + −4, 4 + 0) = (1, 4) = B. ✓

Watch out. It is end minus start. If you swap them you get the vector going the other way, which has the opposite signs.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Subtract the coordinates of the start point from the coordinates of the end point.

1. Write the two points. A column vector from A to B says how far to move across and up to get from A to B.
   A = (0, −3) and B = (−4, −6)
2. Find the across move. Subtract the x-coordinates: end minus start.
   −4 − 0 = −4
3. Find the up move. Subtract the y-coordinates: end minus start.
   −6 − (−3) = −3
4. Write the column vector. Across on top, up underneath.
   →AB = −4
   −3

Check. Start at A and add the vector: (0 + −4, −3 + −3) = (−4, −6) = B. ✓

Watch out. It is end minus start. If you swap them you get the vector going the other way, which has the opposite signs.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Add the vector to the coordinates of the start point.

1. Write the start point and the vector. A column vector is a move: across then up.
   A = (2, 1), →AB = 3
   4
2. Add the across move to the x-coordinate. The top number moves the point right (if positive) or left (if negative).
   2 + 3 = 5
3. Add the up move to the y-coordinate. The bottom number moves the point up (if positive) or down (if negative).
   1 + 4 = 5
4. Write the coordinates.
   B = (5, 5)

Check. Subtract to go back: (5 − 3, 5 − 4) = (2, 1) = A. ✓

Watch out. The top number changes x (sideways) and the bottom number changes y (up and down). Do not swap them.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Add the vector to the coordinates of the start point.

1. Write the start point and the vector. A column vector is a move: across then up.
   A = (−1, 5), →AB = 4
   −2
2. Add the across move to the x-coordinate. The top number moves the point right (if positive) or left (if negative).
   −1 + 4 = 3
3. Add the up move to the y-coordinate. The bottom number moves the point up (if positive) or down (if negative).
   5 + (−2) = 3
4. Write the coordinates.
   B = (3, 3)

Check. Subtract to go back: (3 − 4, 3 − (−2)) = (−1, 5) = A. ✓

Watch out. The top number changes x (sideways) and the bottom number changes y (up and down). Do not swap them.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Add the vector to the coordinates of the start point.

1. Write the start point and the vector. A column vector is a move: across then up.
   A = (0, 0), →AB = −3
   −6
2. Add the across move to the x-coordinate. The top number moves the point right (if positive) or left (if negative).
   0 + (−3) = −3
3. Add the up move to the y-coordinate. The bottom number moves the point up (if positive) or down (if negative).
   0 + (−6) = −6
4. Write the coordinates.
   B = (−3, −6)

Check. Subtract to go back: (−3 − (−3), −6 − (−6)) = (0, 0) = A. ✓

Watch out. The top number changes x (sideways) and the bottom number changes y (up and down). Do not swap them.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Add the vector to the coordinates of the start point.

1. Write the start point and the vector. A column vector is a move: across then up.
   A = (6, −2), →AB = −2
   5
2. Add the across move to the x-coordinate. The top number moves the point right (if positive) or left (if negative).
   6 + (−2) = 4
3. Add the up move to the y-coordinate. The bottom number moves the point up (if positive) or down (if negative).
   −2 + 5 = 3
4. Write the coordinates.
   B = (4, 3)

Check. Subtract to go back: (4 − (−2), 3 − 5) = (6, −2) = A. ✓

Watch out. The top number changes x (sideways) and the bottom number changes y (up and down). Do not swap them.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Multiply the top number and the bottom number of the vector by the scalar.

1. Write the multiplication. A scalar multiple means the vector is stretched (or shrunk) by the number in front.
   −1a = −1 × 3
   −2
2. Multiply the top number. Multiply every number in the column vector by the scalar.
   −1 × 3 = −3
3. Multiply the bottom number. Do the same with the bottom number.
   −1 × (−2) = 2
4. Write the answer. Put the two results in a column.
   −1a = −3
   2

Check. The new vector is −1 times as long as a and points the opposite way. ✓

Watch out. Multiply both numbers, not just the top one. A negative scalar also reverses the direction.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Multiply the top number and the bottom number of the vector by the scalar.

1. Write the multiplication. A scalar multiple means the vector is stretched (or shrunk) by the number in front.
   2a = 2 × 3
   −2
2. Multiply the top number. Multiply every number in the column vector by the scalar.
   2 × 3 = 6
3. Multiply the bottom number. Do the same with the bottom number.
   2 × (−2) = −4
4. Write the answer. Put the two results in a column.
   2a = 6
   −4

Check. The new vector is 2 times as long as a and points the same way. ✓

Watch out. Multiply both numbers, not just the top one.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Multiply the top number and the bottom number of the vector by the scalar.

1. Write the multiplication. A scalar multiple means the vector is stretched (or shrunk) by the number in front.
   −3a = −3 × 3
   −2
2. Multiply the top number. Multiply every number in the column vector by the scalar.
   −3 × 3 = −9
3. Multiply the bottom number. Do the same with the bottom number.
   −3 × (−2) = 6
4. Write the answer. Put the two results in a column.
   −3a = −9
   6

Check. The new vector is −3 times as long as a and points the opposite way. ✓

Watch out. Multiply both numbers, not just the top one. A negative scalar also reverses the direction.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. To add column vectors, add the top numbers and add the bottom numbers separately.

1. Write the two column vectors. Each column vector has a top number (across) and a bottom number (up).
   a + b = 3
   2 + 4
   1
2. Add the top numbers. To add vectors, add the matching numbers: top with top.
   3 + 4 = 7
3. Add the bottom numbers. Now do the same with the bottom with bottom.
   2 + 1 = 3
4. Write the answer as a column vector. Put the new top number over the new bottom number.
   a + b = 7
   3

Check. On a grid: start at the origin, go 3 right and 2 up, then follow b. You arrive 7 right and 3 up of the start. ✓

Watch out. Take care with minus signs: subtracting a negative number adds. For example 3 − (−2) = 5.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. To add column vectors, add the top numbers and add the bottom numbers separately.

1. Write the two column vectors. Each column vector has a top number (across) and a bottom number (up).
   a + b = −1
   5 + 2
   −3
2. Add the top numbers. To add vectors, add the matching numbers: top with top.
   −1 + 2 = 1
3. Add the bottom numbers. Now do the same with the bottom with bottom.
   5 + −3 = 2
4. Write the answer as a column vector. Put the new top number over the new bottom number.
   a + b = 1
   2

Check. On a grid: start at the origin, go 1 left and 5 up, then follow b. You arrive 1 right and 2 up of the start. ✓

Watch out. Take care with minus signs: subtracting a negative number adds. For example 3 − (−2) = 5.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. To add column vectors, add the top numbers and add the bottom numbers separately.

1. Write the two column vectors. Each column vector has a top number (across) and a bottom number (up).
   a + b = 6
   −4 + −2
   −2
2. Add the top numbers. To add vectors, add the matching numbers: top with top.
   6 + −2 = 4
3. Add the bottom numbers. Now do the same with the bottom with bottom.
   −4 + −2 = −6
4. Write the answer as a column vector. Put the new top number over the new bottom number.
   a + b = 4
   −6

Check. On a grid: start at the origin, go 6 right and 4 down, then follow b. You arrive 4 right and 6 down of the start. ✓

Watch out. Take care with minus signs: subtracting a negative number adds. For example 3 − (−2) = 5.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. To add column vectors, add the top numbers and add the bottom numbers separately.

1. Write the two column vectors. Each column vector has a top number (across) and a bottom number (up).
   a + b = 0
   7 + −5
   3
2. Add the top numbers. To add vectors, add the matching numbers: top with top.
   0 + −5 = −5
3. Add the bottom numbers. Now do the same with the bottom with bottom.
   7 + 3 = 10
4. Write the answer as a column vector. Put the new top number over the new bottom number.
   a + b = −5
   10

Check. On a grid: start at the origin, go 0 right and 7 up, then follow b. You arrive 5 left and 10 up of the start. ✓

Watch out. Take care with minus signs: subtracting a negative number adds. For example 3 − (−2) = 5.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. To subtract column vectors, subtract the top numbers and subtract the bottom numbers separately.

1. Write the two column vectors. Each column vector has a top number (across) and a bottom number (up).
   a − b = 5
   7 − 2
   3
2. Subtract the top numbers. To subtract vectors, subtract the matching numbers: top with top.
   5 − 2 = 3
3. Subtract the bottom numbers. Now do the same with the bottom with bottom.
   7 − 3 = 4
4. Write the answer as a column vector. Put the new top number over the new bottom number.
   a − b = 3
   4

Check. On a grid: start at the origin, go 5 right and 7 up, then go the opposite way along b. You arrive 3 right and 4 up of the start. ✓

Watch out. Take care with minus signs: subtracting a negative number adds. For example 3 − (−2) = 5.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. To subtract column vectors, subtract the top numbers and subtract the bottom numbers separately.

1. Write the two column vectors. Each column vector has a top number (across) and a bottom number (up).
   a − b = 1
   −2 − 4
   5
2. Subtract the top numbers. To subtract vectors, subtract the matching numbers: top with top.
   1 − 4 = −3
3. Subtract the bottom numbers. Now do the same with the bottom with bottom.
   −2 − 5 = −7
4. Write the answer as a column vector. Put the new top number over the new bottom number.
   a − b = −3
   −7

Check. On a grid: start at the origin, go 1 right and 2 down, then go the opposite way along b. You arrive 3 left and 7 down of the start. ✓

Watch out. Take care with minus signs: subtracting a negative number adds. For example 3 − (−2) = 5.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. To subtract column vectors, subtract the top numbers and subtract the bottom numbers separately.

1. Write the two column vectors. Each column vector has a top number (across) and a bottom number (up).
   a − b = −3
   6 − −1
   2
2. Subtract the top numbers. To subtract vectors, subtract the matching numbers: top with top.
   −3 − (−1) = −2
3. Subtract the bottom numbers. Now do the same with the bottom with bottom.
   6 − 2 = 4
4. Write the answer as a column vector. Put the new top number over the new bottom number.
   a − b = −2
   4

Check. On a grid: start at the origin, go 3 left and 6 up, then go the opposite way along b. You arrive 2 left and 4 up of the start. ✓

Watch out. Take care with minus signs: subtracting a negative number adds. For example 3 − (−2) = 5.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. To subtract column vectors, subtract the top numbers and subtract the bottom numbers separately.

1. Write the two column vectors. Each column vector has a top number (across) and a bottom number (up).
   a − b = 2
   −4 − −6
   −9
2. Subtract the top numbers. To subtract vectors, subtract the matching numbers: top with top.
   2 − (−6) = 8
3. Subtract the bottom numbers. Now do the same with the bottom with bottom.
   −4 − (−9) = 5
4. Write the answer as a column vector. Put the new top number over the new bottom number.
   a − b = 8
   5

Check. On a grid: start at the origin, go 2 right and 4 down, then go the opposite way along b. You arrive 8 right and 5 up of the start. ✓

Watch out. Take care with minus signs: subtracting a negative number adds. For example 3 − (−2) = 5.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Multiply the top number and the bottom number of the vector by the scalar.

1. Write the multiplication. A scalar multiple means the vector is stretched (or shrunk) by the number in front.
   2a = 2 × 3
   −1
2. Multiply the top number. Multiply every number in the column vector by the scalar.
   2 × 3 = 6
3. Multiply the bottom number. Do the same with the bottom number.
   2 × (−1) = −2
4. Write the answer. Put the two results in a column.
   2a = 6
   −2

Check. The new vector is 2 times as long as a and points the same way. ✓

Watch out. Multiply both numbers, not just the top one.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Multiply the top number and the bottom number of the vector by the scalar.

1. Write the multiplication. A scalar multiple means the vector is stretched (or shrunk) by the number in front.
   −3a = −3 × 2
   4
2. Multiply the top number. Multiply every number in the column vector by the scalar.
   −3 × 2 = −6
3. Multiply the bottom number. Do the same with the bottom number.
   −3 × 4 = −12
4. Write the answer. Put the two results in a column.
   −3a = −6
   −12

Check. The new vector is −3 times as long as a and points the opposite way. ✓

Watch out. Multiply both numbers, not just the top one. A negative scalar also reverses the direction.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Multiply the top number and the bottom number of the vector by the scalar.

1. Write the multiplication. A scalar multiple means the vector is stretched (or shrunk) by the number in front.
   0.5a = 0.5 × 8
   −6
2. Multiply the top number. Multiply every number in the column vector by the scalar.
   0.5 × 8 = 4
3. Multiply the bottom number. Do the same with the bottom number.
   0.5 × (−6) = −3
4. Write the answer. Put the two results in a column.
   0.5a = 4
   −3

Check. The new vector is 0.5 times as long as a and points the same way. ✓

Watch out. Multiply both numbers, not just the top one.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Multiply the top number and the bottom number of the vector by the scalar.

1. Write the multiplication. A scalar multiple means the vector is stretched (or shrunk) by the number in front.
   0.25a = 0.25 × 12
   8
2. Multiply the top number. Multiply every number in the column vector by the scalar.
   0.25 × 12 = 3
3. Multiply the bottom number. Do the same with the bottom number.
   0.25 × 8 = 2
4. Write the answer. Put the two results in a column.
   0.25a = 3
   2

Check. The new vector is 0.25 times as long as a and points the same way. ✓

Watch out. Multiply both numbers, not just the top one.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Multiply each vector by its scalar first, then add or subtract the top numbers and the bottom numbers.

1. Write each vector as a column. We can only add and subtract once everything is in column form.
   a + b = 2
   3 + −1
   4
2. Do the scalar multiplications first. Multiply every number in each vector by its scalar (like BIDMAS: multiply before adding).
   2
   3 + −1
   4
3. Add the top numbers. Add the matching numbers: top with top.
   2 + (−1) = 1
4. Add the bottom numbers. Then bottom with bottom.
   3 + 4 = 7
5. Write the answer.
   a + b = 1
   7

Check. Check the top numbers again: 2 −1 = 1. ✓

Watch out. Do not forget to multiply both numbers, and watch the signs when you subtract a vector.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Multiply each vector by its scalar first, then add or subtract the top numbers and the bottom numbers.

1. Write each vector as a column. We can only add and subtract once everything is in column form.
   2a + b = 2 × 2
   3 + −1
   4
2. Do the scalar multiplications first. Multiply every number in each vector by its scalar (like BIDMAS: multiply before adding).
   4
   6 + −1
   4
3. Add the top numbers. Add the matching numbers: top with top.
   4 + (−1) = 3
4. Add the bottom numbers. Then bottom with bottom.
   6 + 4 = 10
5. Write the answer.
   2a + b = 3
   10

Check. Check the top numbers again: 4 −1 = 3. ✓

Watch out. Do not forget to multiply both numbers, and watch the signs when you subtract a vector.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Multiply each vector by its scalar first, then add or subtract the top numbers and the bottom numbers.

1. Write each vector as a column. We can only add and subtract once everything is in column form.
   3a − 2b = 3 × 2
   3 + −2 × −1
   4
2. Do the scalar multiplications first. Multiply every number in each vector by its scalar (like BIDMAS: multiply before adding).
   6
   9 + 2
   −8
3. Add the top numbers. Add the matching numbers: top with top.
   6 + 2 = 8
4. Add the bottom numbers. Then bottom with bottom.
   9 + (−8) = 1
5. Write the answer.
   3a − 2b = 8
   1

Check. Check the top numbers again: 6 + 2 = 8. ✓

Watch out. Do not forget to multiply both numbers, and watch the signs when you subtract a vector.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Multiply each vector by its scalar first, then add or subtract the top numbers and the bottom numbers.

1. Write each vector as a column. We can only add and subtract once everything is in column form.
   a − 3b = 2
   3 + −3 × −1
   4
2. Do the scalar multiplications first. Multiply every number in each vector by its scalar (like BIDMAS: multiply before adding).
   2
   3 + 3
   −12
3. Add the top numbers. Add the matching numbers: top with top.
   2 + 3 = 5
4. Add the bottom numbers. Then bottom with bottom.
   3 + (−12) = −9
5. Write the answer.
   a − 3b = 5
   −9

Check. Check the top numbers again: 2 + 3 = 5. ✓

Watch out. Do not forget to multiply both numbers, and watch the signs when you subtract a vector.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Two vectors are parallel when one is a scalar multiple of the other. Find the multiplier for the top numbers and the bottom numbers and see if it matches.

1. Compare the top numbers. If the vectors are parallel, one is a multiple of the other, so the top numbers have the same scale factor as the bottom numbers.
   1 ÷ 3 = 0.333333333333333333333333333333333333333333333333333333333333
2. Compare the bottom numbers. Is the multiplier the same?
   2 ÷ 6 = 0.333333333333333333333333333333333333333333333333333333333333
3. Write the conclusion. Both numbers are multiplied by the same scalar.
   b = 0.333333333333333333333333333333333333333333333333333333333333a, so they are parallel.

Check. 0.333333333333333333333333333333333333333333333333333333333333 × 3
6 = 1
2. ✓

Watch out. The scalar can be negative. Vectors that point in opposite directions along the same line are still parallel.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Two vectors are parallel when one is a scalar multiple of the other. Find the multiplier for the top numbers and the bottom numbers and see if it matches.

1. Compare the top numbers. If the vectors are parallel, one is a multiple of the other, so the top numbers have the same scale factor as the bottom numbers.
   −4 ÷ 2 = −2
2. Compare the bottom numbers. Is the multiplier the same?
   6 ÷ −3 = −2
3. Write the conclusion. Both numbers are multiplied by the same scalar.
   b = −2a, so they are parallel.

Check. −2 × 2
−3 = −4
6. ✓

Watch out. The scalar can be negative. Vectors that point in opposite directions along the same line are still parallel.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Two vectors are parallel when one is a scalar multiple of the other. Find the multiplier for the top numbers and the bottom numbers and see if it matches.

1. Compare the top numbers. If the vectors are parallel, both numbers are multiplied by the same scalar.
   8 ÷ 4 = 2
2. Compare the bottom numbers. Is the multiplier the same?
   3 ÷ 1 = 3
3. Write the conclusion. The multipliers are different, so one vector is not a scalar multiple of the other.
   The vectors are not parallel.

Check. No single number turns 4
1 into 8
3. ✓

Watch out. The scalar can be negative. Vectors that point in opposite directions along the same line are still parallel.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. Two vectors are parallel when one is a scalar multiple of the other. Find the multiplier for the top numbers and the bottom numbers and see if it matches.

1. Compare the top numbers. If the vectors are parallel, one is a multiple of the other, so the top numbers have the same scale factor as the bottom numbers.
   −10 ÷ 5 = −2
2. Compare the bottom numbers. Is the multiplier the same?
   The bottom numbers are both 0.
3. Write the conclusion. Both numbers are multiplied by the same scalar.
   b = −2a, so they are parallel.

Check. −2 × 5
0 = −10
0. ✓

Watch out. The scalar can be negative. Vectors that point in opposite directions along the same line are still parallel.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. The magnitude (length) of a vector is found with Pythagoras’ theorem on its two numbers.

1. Draw the right-angled triangle. The vector a goes 3 across and 4 up or down. These two moves are at right angles, so the length of the vector is the hypotenuse.
   Shorter sides: 3 and 4
2. Use Pythagoras’ theorem. Square each number, add them, then take the square root. Squaring removes any minus signs.
   |a|² = 3² + 4² = 9 + 16 = 25
3. Square root. Find the length.
   |a| = √25 = 5

Check. The length must be at least as big as each number: 5 = 5.00 is bigger than 3 and 4. ✓

Watch out. Write |a| for the magnitude. The magnitude is never negative, even if the numbers in the vector are.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. The magnitude (length) of a vector is found with Pythagoras’ theorem on its two numbers.

1. Draw the right-angled triangle. The vector a goes 5 across and 12 up or down. These two moves are at right angles, so the length of the vector is the hypotenuse.
   Shorter sides: 5 and 12
2. Use Pythagoras’ theorem. Square each number, add them, then take the square root. Squaring removes any minus signs.
   |a|² = 5² + 12² = 25 + 144 = 169
3. Square root. Find the length.
   |a| = √169 = 13

Check. The length must be at least as big as each number: 13 = 13.00 is bigger than 5 and 12. ✓

Watch out. Write |a| for the magnitude. The magnitude is never negative, even if the numbers in the vector are.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. The magnitude (length) of a vector is found with Pythagoras’ theorem on its two numbers.

1. Draw the right-angled triangle. The vector a goes 8 across and 6 up or down. These two moves are at right angles, so the length of the vector is the hypotenuse.
   Shorter sides: 8 and 6
2. Use Pythagoras’ theorem. Square each number, add them, then take the square root. Squaring removes any minus signs.
   |a|² = 8² + 6² = 64 + 36 = 100
3. Square root. Find the length.
   |a| = √100 = 10

Check. The length must be at least as big as each number: 10 = 10.00 is bigger than 8 and 6. ✓

Watch out. Write |a| for the magnitude. The magnitude is never negative, even if the numbers in the vector are.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. The magnitude (length) of a vector is found with Pythagoras’ theorem on its two numbers.

1. Draw the right-angled triangle. The vector a goes 7 across and 24 up or down. These two moves are at right angles, so the length of the vector is the hypotenuse.
   Shorter sides: 7 and 24
2. Use Pythagoras’ theorem. Square each number, add them, then take the square root. Squaring removes any minus signs.
   |a|² = (−7)² + 24² = 49 + 576 = 625
3. Square root. Find the length.
   |a| = √625 = 25

Check. The length must be at least as big as each number: 25 = 25.00 is bigger than 7 and 24. ✓

Watch out. Write |a| for the magnitude. The magnitude is never negative, even if the numbers in the vector are.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. The magnitude (length) of a vector is found with Pythagoras’ theorem on its two numbers.

1. Draw the right-angled triangle. The vector a goes 2 across and 5 up or down. These two moves are at right angles, so the length of the vector is the hypotenuse.
   Shorter sides: 2 and 5
2. Use Pythagoras’ theorem. Square each number, add them, then take the square root. Squaring removes any minus signs.
   |a|² = 2² + 5² = 4 + 25 = 29
3. Square root and round. Use a calculator and round as the question asks.
   |a| = √29 = 5.3852… = 5.4

Check. The length must be at least as big as each number: 5.4 ≈ 5.39 is bigger than 2 and 5. ✓

Watch out. Write |a| for the magnitude. The magnitude is never negative, even if the numbers in the vector are.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. The magnitude (length) of a vector is found with Pythagoras’ theorem on its two numbers.

1. Draw the right-angled triangle. The vector a goes 3 across and 7 up or down. These two moves are at right angles, so the length of the vector is the hypotenuse.
   Shorter sides: 3 and 7
2. Use Pythagoras’ theorem. Square each number, add them, then take the square root. Squaring removes any minus signs.
   |a|² = (−3)² + (−7)² = 9 + 49 = 58
3. Square root and round. Use a calculator and round as the question asks.
   |a| = √58 = 7.6158… = 7.6

Check. The length must be at least as big as each number: 7.6 ≈ 7.62 is bigger than 3 and 7. ✓

Watch out. Write |a| for the magnitude. The magnitude is never negative, even if the numbers in the vector are.

Key words. A vector has a size (its length) and a direction. A column vector writes it as two numbers: how far across (right is +) over how far up (up is +). A scalar is an ordinary number.

Method. The magnitude (length) of a vector is found with Pythagoras’ theorem on its two numbers.

1. Draw the right-angled triangle. The vector a goes 6 across and 1 up or down. These two moves are at right angles, so the length of the vector is the hypotenuse.
   Shorter sides: 6 and 1
2. Use Pythagoras’ theorem. Square each number, add them, then take the square root. Squaring removes any minus signs.
   |a|² = 6² + (−1)² = 36 + 1 = 37
3. Square root and round. Use a calculator and round as the question asks.
   |a| = √37 = 6.0828… = 6.1

Check. The length must be at least as big as each number: 6.1 ≈ 6.08 is bigger than 6 and 1. ✓

Watch out. Write |a| for the magnitude. The magnitude is never negative, even if the numbers in the vector are.