High School Maths · Year 2 · Unit 12 · Lesson 1 of 5

Column Vectors: The Arrows That Tell You Exactly Where to Go

About 15 minutes · column vectors and what the top and bottom numbers mean · the vector between two points · adding, subtracting and multiplying by a number (a scalar) · equal, negative and parallel vectors · the magnitude |a| by Pythagoras, as a whole number or an exact surd · solving vector equations · no calculator · exit quiz

Hook

A delivery drone flies 51, then 17.

Which single move takes it straight from the start to the finish, and how far is that?

Add the tops and add the bottoms: 5 + 1 = 6 across and 1 + 7 = 8 up, so the single move is 68. In Year 1 you used Pythagoras for distances: √(6² + 8²) = √100 = 10 squares. The arrow that does both moves at once is a vector. This lesson is all about adding, scaling and measuring them.

Learn

Master the essentials

Equal vectors have the same length and direction, wherever they are drawn.

Column vector

The top number moves right (+) or left (−). The bottom number moves up (+) or down (−).

3−2 means 3 right and 2 down

Naming vectors

Use a bold letter such as a, or the two end points with an arrow: AB goes from A to B. Work out end minus start.

A(1, 2) to B(4, 7): AB = 35

Adding

Add the tops, then add the bottoms. On a diagram, put the arrows nose to tail.

2−5 + 43 = 6−2

Subtracting

Take away the tops, then the bottoms. Watch the signs: taking away a negative adds.

71 − 2−4 = 55

Scalar multiple

Multiply both numbers by the scalar (an ordinary number). The arrow stretches but keeps its direction.

32−1 = 6−3

Negative vector

−a has the same length as a but points the opposite way. So BA = −AB.

−4−3 = −43

Parallel vectors

Two vectors are parallel when one is a multiple of the other.

8−12 = 42−3

Magnitude

The length |a| comes from Pythagoras: square both numbers, add, then take the square root. Simplify any surd.

|5−12| = √169 = 13, |24| = √20 = 2√5

Worked examples

See it done

1. a = 3−2 and b = −14. Find 2a − 3b as a column vector.

  1. Multiply first: 2a = 6−4 and 3b = −312.
  2. Subtract the tops: 6 − (−3) = 6 + 3 = 9.
  3. Subtract the bottoms: −4 − 12 = −16.

2a − 3b = 9−16

2. Find the magnitude of −64. Give your answer as a surd in its simplest form.

  1. Square both numbers: (−6)² = 36 and 4² = 16.
  2. Add: 36 + 16 = 52, so the magnitude is √52.
  3. Simplify: 52 = 4 × 13, so √52 = √4 × √13 = 2√13.

2√13

3. Find x and y when x5 + 23y = 1−3.

  1. Top: x + 2 × 3 = 1, so x + 6 = 1 and x = −5.
  2. Bottom: 5 + 2y = −3, so 2y = −8 and y = −4.
  3. Check: −55 + 6−8 = 1−3 ✓.

x = −5, y = −4

Common mistake: Slipping on the signs. 4−1 − −23 is 6−4, because 4 − (−2) = 6. And a magnitude is never found by subtracting squares: |3−4| = √(9 + 16) = 5, because (−4)² = +16.

PractiseNo calculator

10 questions with instant feedback

Choose the correct answer. Work out the tops and the bottoms separately, and write each sign down before you calculate. Questions 1–3 are warm-ups, 4–7 are at exam standard and 8–10 are challenge questions. After you answer, the feedback shows the full solution.

Score: 0 / 10Answered: 0 / 10

Warm-up (The basic skill: one or two steps.)

1.a = 5−3 and b = −26. Find a + b.

Correct.

Solution

  1. Add the tops: 5 + (−2) = 3.
  2. Add the bottoms: −3 + 6 = 3.
  3. So a + b = 33.

Not quite. This works out a − b. To add, add the tops and add the bottoms.

Solution

  1. Add the tops: 5 + (−2) = 3.
  2. Add the bottoms: −3 + 6 = 3.
  3. So a + b = 33.

Answer: A. 33.

Not quite. This drops the minus sign on −3. The bottom is −3 + 6 = 3.

Solution

  1. Add the tops: 5 + (−2) = 3.
  2. Add the bottoms: −3 + 6 = 3.
  3. So a + b = 33.

Answer: A. 33.

Not quite. This multiplies the parts. To add vectors, add them.

Solution

  1. Add the tops: 5 + (−2) = 3.
  2. Add the bottoms: −3 + 6 = 3.
  3. So a + b = 33.

Answer: A. 33.

2.Find the magnitude of −129.

Correct.

Solution

  1. Square both numbers: (−12)² = 144 and 9² = 81.
  2. Add: 144 + 81 = 225.
  3. Square root: √225 = 15.

Not quite. This adds the sizes of the two numbers. Use Pythagoras.

Solution

  1. Square both numbers: (−12)² = 144 and 9² = 81.
  2. Add: 144 + 81 = 225.
  3. Square root: √225 = 15.

Answer: A. 15.

Not quite. This forgets the square root at the end: √225 = 15.

Solution

  1. Square both numbers: (−12)² = 144 and 9² = 81.
  2. Add: 144 + 81 = 225.
  3. Square root: √225 = 15.

Answer: A. 15.

Not quite. This subtracts the squares. (−12)² and 9² are both positive, and you add them.

Solution

  1. Square both numbers: (−12)² = 144 and 9² = 81.
  2. Add: 144 + 81 = 225.
  3. Square root: √225 = 15.

Answer: A. 15.

3.Point A is at (−3, 2) and point B is at (4, −1). Write AB as a column vector.

Not quite. This is the vector from B to A. Use end minus start: B − A.

Solution

  1. End minus start: B − A.
  2. Top: 4 − (−3) = 7.
  3. Bottom: −1 − 2 = −3.

Answer: D. 7−3.

Not quite. This adds the coordinates. Subtract: end minus start.

Solution

  1. End minus start: B − A.
  2. Top: 4 − (−3) = 7.
  3. Bottom: −1 − 2 = −3.

Answer: D. 7−3.

Not quite. The bottom number is worked out the wrong way round. It is −1 − 2 = −3.

Solution

  1. End minus start: B − A.
  2. Top: 4 − (−3) = 7.
  3. Bottom: −1 − 2 = −3.

Answer: D. 7−3.

Correct.

Solution

  1. End minus start: B − A.
  2. Top: 4 − (−3) = 7.
  3. Bottom: −1 − 2 = −3.

Exam standard (Several steps, like real exam questions.)

4.p = 4−3 and q = −25. Find 3p − 2q.

Not quite. This adds 2q. The question says take it away.

Solution

  1. 3p = 12−9 and 2q = −410.
  2. Tops: 12 − (−4) = 16.
  3. Bottoms: −9 − 10 = −19.

Answer: C. 16−19.

Not quite. This adds 10 at the bottom. −9 − 10 = −19.

Solution

  1. 3p = 12−9 and 2q = −410.
  2. Tops: 12 − (−4) = 16.
  3. Bottoms: −9 − 10 = −19.

Answer: C. 16−19.

Correct.

Solution

  1. 3p = 12−9 and 2q = −410.
  2. Tops: 12 − (−4) = 16.
  3. Bottoms: −9 − 10 = −19.

Not quite. This forgets to multiply q by 2.

Solution

  1. 3p = 12−9 and 2q = −410.
  2. Tops: 12 − (−4) = 16.
  3. Bottoms: −9 − 10 = −19.

Answer: C. 16−19.

5.a = 24 and b = −25. Find |a − 2b|. Give your answer as a surd in its simplest form.

Not quite. This works out a + 2b. The question says take away 2b.

Solution

  1. a − 2b = 24 − −410 = 6−6.
  2. Magnitude: √(6² + (−6)²) = √(36 + 36) = √72.
  3. 72 = 36 × 2, so √72 = 6√2.

Answer: C. 6√2.

Not quite. This adds the parts 6 + 6. Use Pythagoras.

Solution

  1. a − 2b = 24 − −410 = 6−6.
  2. Magnitude: √(6² + (−6)²) = √(36 + 36) = √72.
  3. 72 = 36 × 2, so √72 = 6√2.

Answer: C. 6√2.

Correct.

Solution

  1. a − 2b = 24 − −410 = 6−6.
  2. Magnitude: √(6² + (−6)²) = √(36 + 36) = √72.
  3. 72 = 36 × 2, so √72 = 6√2.

Not quite. This adds the parts without squaring them: √(6 + 6).

Solution

  1. a − 2b = 24 − −410 = 6−6.
  2. Magnitude: √(6² + (−6)²) = √(36 + 36) = √72.
  3. 72 = 36 × 2, so √72 = 6√2.

Answer: C. 6√2.

6.The table shows three vectors. Find u + 2v − w as a column vector.

Vectoruvw
Column vector6−1−234−5

Not quite. This adds w. The question says − w.

Solution

  1. 2v = −46.
  2. Tops: 6 + (−4) − 4 = −2.
  3. Bottoms: −1 + 6 − (−5) = −1 + 6 + 5 = 10.

Answer: C. −210.

Not quite. This forgets to double v.

Solution

  1. 2v = −46.
  2. Tops: 6 + (−4) − 4 = −2.
  3. Bottoms: −1 + 6 − (−5) = −1 + 6 + 5 = 10.

Answer: C. −210.

Correct.

Solution

  1. 2v = −46.
  2. Tops: 6 + (−4) − 4 = −2.
  3. Bottoms: −1 + 6 − (−5) = −1 + 6 + 5 = 10.

Not quite. This takes away 5 at the bottom. Taking away −5 means adding 5.

Solution

  1. 2v = −46.
  2. Tops: 6 + (−4) − 4 = −2.
  3. Bottoms: −1 + 6 − (−5) = −1 + 6 + 5 = 10.

Answer: C. −210.

7.Find x and y when 2x3 − 5y = 7−4.

Not quite. From −y = −10, divide both sides by −1: y = 10.

Solution

  1. Top: 2x − 5 = 7, so 2x = 12 and x = 6.
  2. Bottom: 6 − y = −4, so −y = −10 and y = 10.
  3. Check: 126 − 510 = 7−4 ✓.

Answer: B. x = 6, y = 10.

Correct.

Solution

  1. Top: 2x − 5 = 7, so 2x = 12 and x = 6.
  2. Bottom: 6 − y = −4, so −y = −10 and y = 10.
  3. Check: 126 − 510 = 7−4 ✓.

Not quite. 2x − 5 = 7 gives 2x = 12. Add the 5; do not subtract it.

Solution

  1. Top: 2x − 5 = 7, so 2x = 12 and x = 6.
  2. Bottom: 6 − y = −4, so −y = −10 and y = 10.
  3. Check: 126 − 510 = 7−4 ✓.

Answer: B. x = 6, y = 10.

Not quite. This forgets to double the 3. The bottom is 2 × 3 − y = 6 − y.

Solution

  1. Top: 2x − 5 = 7, so 2x = 12 and x = 6.
  2. Bottom: 6 − y = −4, so −y = −10 and y = 10.
  3. Check: 126 − 510 = 7−4 ✓.

Answer: B. x = 6, y = 10.

Challenge (Harder problems that mix ideas or need reasoning.)

8.a = 6−4 and b = k10. a and b are parallel. Find k.

Not quite. This misses the minus sign. The bottoms go from −4 to 10, so the multiplier is negative.

Solution

  1. Parallel, so b is a multiple of a.
  2. The bottoms: 10 ÷ (−4) = −2.5, so the multiplier is −2.5.
  3. k = 6 × (−2.5) = −15.

Answer: B. −15.

Correct.

Solution

  1. Parallel, so b is a multiple of a.
  2. The bottoms: 10 ÷ (−4) = −2.5, so the multiplier is −2.5.
  3. k = 6 × (−2.5) = −15.

Not quite. This divides the wrong way: it uses −4 ÷ 10 instead of 10 ÷ (−4).

Solution

  1. Parallel, so b is a multiple of a.
  2. The bottoms: 10 ÷ (−4) = −2.5, so the multiplier is −2.5.
  3. k = 6 × (−2.5) = −15.

Answer: B. −15.

Not quite. This makes the differences equal: 10 − (−4) = 14 and 6 + 14 = 20. Parallel vectors are multiples.

Solution

  1. Parallel, so b is a multiple of a.
  2. The bottoms: 10 ÷ (−4) = −2.5, so the multiplier is −2.5.
  3. k = 6 × (−2.5) = −15.

Answer: B. −15.

9.a = 2−11 and b = −105. Which working shows that a and b have the same magnitude?

Not quite. This adds the parts. A magnitude needs Pythagoras.

Solution

  1. |a|² = 2² + (−11)² = 4 + 121 = 125.
  2. |b|² = (−10)² + 5² = 100 + 25 = 125.
  3. The same, so both magnitudes are √125 = 5√5 ✓.

Answer: D. |a|² = 2² + (−11)² = 125 and |b|² = (−10)² + 5² = 125, so both are √125 = 5√5.

Not quite. (−11)² = +121. Squares are added, never subtracted.

Solution

  1. |a|² = 2² + (−11)² = 4 + 121 = 125.
  2. |b|² = (−10)² + 5² = 100 + 25 = 125.
  3. The same, so both magnitudes are √125 = 5√5 ✓.

Answer: D. |a|² = 2² + (−11)² = 125 and |b|² = (−10)² + 5² = 125, so both are √125 = 5√5.

Not quite. (−10)² = +100. Squaring a negative number gives a positive.

Solution

  1. |a|² = 2² + (−11)² = 4 + 121 = 125.
  2. |b|² = (−10)² + 5² = 100 + 25 = 125.
  3. The same, so both magnitudes are √125 = 5√5 ✓.

Answer: D. |a|² = 2² + (−11)² = 125 and |b|² = (−10)² + 5² = 125, so both are √125 = 5√5.

Correct.

Solution

  1. |a|² = 2² + (−11)² = 4 + 121 = 125.
  2. |b|² = (−10)² + 5² = 100 + 25 = 125.
  3. The same, so both magnitudes are √125 = 5√5 ✓.

10.P is the point (2, −3). PQ = 54 and QR = −72. Find the coordinates of R and the length of PR.

Not quite. −26 is the vector PR. Add it to P to find the point R.

Solution

  1. PR = 54 + −72 = −26.
  2. R = (2 + (−2), −3 + 6) = (0, 3).
  3. Length: √((−2)² + 6²) = √40 = 2√10.

Answer: D. R(0, 3) and |PR| = 2√10.

Not quite. This adds the parts 2 + 6. The length needs Pythagoras.

Solution

  1. PR = 54 + −72 = −26.
  2. R = (2 + (−2), −3 + 6) = (0, 3).
  3. Length: √((−2)² + 6²) = √40 = 2√10.

Answer: D. R(0, 3) and |PR| = 2√10.

Not quite. This is the distance of R from the origin. The question asks for PR.

Solution

  1. PR = 54 + −72 = −26.
  2. R = (2 + (−2), −3 + 6) = (0, 3).
  3. Length: √((−2)² + 6²) = √40 = 2√10.

Answer: D. R(0, 3) and |PR| = 2√10.

Correct.

Solution

  1. PR = 54 + −72 = −26.
  2. R = (2 + (−2), −3 + 6) = (0, 3).
  3. Length: √((−2)² + 6²) = √40 = 2√10.

Apply

Steer the robot

A robot starts at the origin O. Button A moves it by a = 31. Button B moves it by b = −12.

1. The robot obeys A twice and B three times. Where does it end up?

2. How far is it from O now? Give an exact answer.

3. How many presses of each button take the robot from O to the point (11, 6)?

  1. 2a + 3b = 62 + −36 = 38, so the robot is at (3, 8).
  1. Distance = √(3² + 8²) = √(9 + 64) = √73. (73 has no square factor, so this is its simplest form.)
  1. Let A be pressed m times and B n times: 3m − n = 11 and m + 2n = 6.
  2. From the first, n = 3m − 11. Then m + 6m − 22 = 6, so 7m = 28.
  3. A 4 times and B once. Check: 431 + −12 = 116 ✓.

Next: the Lesson 1 exit quiz. Five questions, recorded on your dashboard. Aim for 4 out of 5.

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