Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From lesson 4: find the vector from (1, 2) to (6, 5).
Answer
(5, 3). Subtract the start from the end. - From Year 8: work out 3 × (−4) + 2 × 5.
Answer
−2. −12 + 10. - From Year 10: a translation moves a shape by (2, −3). What is the effect of doing it twice?
Answer
A translation by (4, −6).
Words used in this lesson
- scalar
- a number that multiplies a vector. For example: the 3 in 3a.
- parallel
- pointing in the same or opposite direction. For example: one is a multiple of the other.
- sum
- the vector found by adding two vectors. For example: a movement then another.
- negative vector
- a vector with the same size and the opposite direction. For example: the reverse journey.
- tip to tail
- joining vectors end to start so they follow one another. For example: a route in two legs.
Do two drones that fly the same legs in opposite orders arrive together?
In a delivery trial near Medellin, one drone flies the leg 2
1 kilometres and then the leg 1
3. A second
drone flies the same two legs but in the opposite order. Each leg is a vector, and two movements one after the other are added.
Writing a bold letter such as a for a vector, we write a + b for doing
a then b.
Predict first: do the two drones end at the same point? Then use the figure, which draws both orders, and see what a negative multiple does.
m = 2, n = −1: m a = (4, 2), n b = (−1, −3), so m a + n b = (3, −1).
Show me the steps
- m = 1, n = 1: a + b is (2, 1) + (1, 3) = (3, 4), and following a then b ends at the same point as b then a.
- m = 2, n = 0: 2a is (4, 2), the same direction as a and twice as long.
- m = −1, n = 1: b − a is (1, 3) − (2, 1) = (−1, 2), so subtracting a is adding the reversed vector.
- Any m and n: the entries are m times 2 plus n times 1 across and m times 1 plus n times 3 up.
Add and subtract the entries. p
q + r
s = p + r
q + s. A scalar
multiple multiplies every entry: kp
q = kp
kq.
Geometry of the rules. Joining the movements tip to tail gives the sum, so AB + BC = AC. Multiplying by a positive number stretches the vector without turning it, and a negative number reverses it. Two vectors are parallel exactly when one is a scalar multiple of the other.
Two common slips. Subtracting a vector by subtracting only some entries, or changing the sign of only one of them: a − b changes the sign of both entries of b. And forgetting that a negative scalar reverses the direction: −2a is twice as long as a and points the opposite way.
Worked examples
1. Given a = 3
2 and b = −1
4, find a + b.
- Add the top entries: 3 + (−1) = 2. Add the bottom entries: 2 + 4 = 6.
- a + b = 2
6.
2. With the same vectors, find 2a − b.
- 2a = 6
4. Subtracting b changes both signs of b: 6 − (−1) = 7 and 4 − 4 = 0. - 2a − b = 7
0.
3. Is 6
−9 parallel to 2
−3?
- Divide each entry of the first by the matching entry of the second: 6 ÷ 2 = 3 and −9 ÷ −3 = 3.
- Both give 3, so the first is 3 times the second, and they are parallel and point the same way.
Your go. The first step is given. Given p = 1
5 and q = 4
−2, find 3p − 2q.
- Find 3p: multiply both entries of p by 3.
- Find 2q, then subtract it entry by entry.
Check the answer
3p = 3
15 and 2q = 8
−4. Subtracting: 3 − 8 = −5 and
15 − (−4) = 19, so 3p − 2q = −5
19.
Check before you go on
1. a = 3
−2 and b = 1
4. Which is a − b?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. p = 4
−1 and q = −2
5. What is the bottom entry of p + q?
3. a = 2
5 and b = −1
4. What is the top entry of 3a − 2b?
Level 2 · Building
4. In a survey, AB = 5
−3 and BC = −2
−6. What is the bottom entry of AC?
5. k2
−3 = 10
−15. What is k?
Level 3 · Stretch
6. The vector k
6 is parallel to −6
9. What is k?
7. The vector 6
−2 + t1
3 has its top entry equal to its bottom entry. What is t?
Something to think about
The vector a + b is the diagonal of the parallelogram built on a and b. What is the other diagonal, and what does it show about a − b?
One way to think about it
The other diagonal joins the tips of a and b, so it is the vector from the tip of a to the tip of b, which is b − a. Reversed, it is a − b. Both diagonals come from the same two sides of the parallelogram.
Exit ticket
8. AB = a and BC = b. Which expression gives the vector from C to A?