This set has 11 original questions on the skills in vectors and transformations. They run from easy to harder. Write out every step on paper before you open an answer, and do not use a calculator unless you need to check.
Coordinates are written P(3, −1) and column vectors as (3, −1), with the upper number first. For a reminder of the notation, see describing a translation with a column vector.
Questions
1. Point P(3, −2) is translated by the vector (−5, 4). Find the image of P.
Show answer
x: 3 + (−5) = −2. y: −2 + 4 = 2.
P′ = (−2, 2)
Check: image minus object = (−2 − 3, 2 − (−2)) = (−5, 4). Correct.
2. A translation maps A(−1, 4) to A′(6, −2). Write its column vector.
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x: 6 − (−1) = 7. y: −2 − 4 = −6.
Vector (7, −6).
Check: −1 + 7 = 6 and 4 + (−6) = −2.
3. a = (2, −3) and b = (−4, 1). Find a + b.
Show answer
Upper: 2 + (−4) = −2. Lower: −3 + 1 = −2.
a + b = (−2, −2)
4. Using the same a and b, find 2a − b.
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2a = (4, −6). Subtract b: (4 − (−4), −6 − 1) = (8, −7).
2a − b = (8, −7)
5. AB = (3, 4) and BC = (−7, 2). (a) Find AC. (b) Find the length of AB.
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(a) AC = AB + BC = (3 + (−7), 4 + 2) = (−4, 6).
(b) Length of AB = √(3² + 4²) = √(9 + 16) = √25 = 5.
6. OA = (3, −1) and OB = (−2, 5). (a) Find AB. (b) Find the position vector of M, the midpoint of AB.
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(a) AB = OB − OA = (−2 − 3, 5 − (−1)) = (−5, 6).
(b) OM = OA + ½AB = (3 + (−2.5), −1 + 3) = (0.5, 2).
Check: halve the sum of the position vectors: (3 + (−2), −1 + 5) ÷ 2 = (1, 4) ÷ 2 = (0.5, 2).
7. A(−3, 2) and B(5, −6). P lies on AB with AP:PB = 1:3. Find the coordinates of P.
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AB = (5 − (−3), −6 − 2) = (8, −8). Total parts = 1 + 3 = 4, so AP = ¼AB = (2, −2).
P = (−3 + 2, 2 + (−2)) = (−1, 0).
Check: PB = (5 − (−1), −6 − 0) = (6, −6), which is three times AP = (2, −2). Correct.
8. A(1, −4) and B(11, 6). P lies on AB with AP:PB = 3:2. Find the coordinates of P.
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AB = (10, 10). Total parts = 5, so AP = ⅗AB = (6, 6).
P = (1 + 6, −4 + 6) = (7, 2).
Check: PB = (11 − 7, 6 − 2) = (4, 4). AP = (6, 6) is 3 parts of (2, 2) and PB is 2 parts. Correct.
9. (a) Reflect (−3, 4) in the line x = 1. (b) Reflect (2, 5) in the line y = x.
Show answer
(a) (−3, 4) is 1 − (−3) = 4 to the left of the line. The image is 4 to the right: 1 + 4 = 5. The y value stays 4. (5, 4).
(b) Reflection in y = x swaps the coordinates: (5, 2).
10. Rotate (−2, 3) through 90° anticlockwise about the origin.
Show answer
A 90° anticlockwise turn maps (x, y) to (−y, x). So (−2, 3) becomes (−3, −2).
(−3, −2)
Check: both points are √13 from the origin, and the point moves from the second quadrant to the third, which is anticlockwise.
11. The point P(3, 4) is mapped to P′(−1, −2) by a rotation through 180°. Find the centre of rotation, and describe the transformation completely.
Show answer
In a half turn, the centre is the midpoint of P and P′: ((3 + (−1)) ÷ 2, (4 + (−2)) ÷ 2) = (1, 1).
Rotation through 180° about the point (1, 1).
Check: image = 2 × centre − point = (2 − 3, 2 − 4) = (−1, −2). Correct. A half turn needs no direction.
If you got these wrong
| What went wrong | Go to |
|---|---|
| Image or vector signs wrong (questions 1 to 2) | Describe a translation using a column vector |
| Added or reversed vectors wrongly (questions 3 to 5) | Add displacement vectors on a diagram |
| Ratio point wrong, often half instead of a third (questions 7 to 8) | Find a point using a vector ratio |
| Reflection or rotation wrong or incomplete (questions 9 to 11) | Describe a reflection or rotation completely |
| Mixed up OA, OB and AB (question 6) | Distinguish position from displacement |
Put each slip into the mistake log and retest queue and retry a fresh version a few days later. The non-calculator working trainer gives extra arithmetic practice.
If one error type keeps returning, our teachers can trace it with you in online one-to-one Mathematics tuition.