These eight questions use new values and get harder, from vector arithmetic to direction angles and unit vectors. Write your own working first, then open the answer.
They follow the lessons in vectors. Give decimal answers to three significant figures and angles to one decimal place.
Questions
Question 1. Given a = 4i − j and b = −2i + 5j, find 3a + 2b.
Answer
3a = 12i − 3j and 2b = −4i + 10j. Adding gives 3a + 2b = (12 − 4)i + (−3 + 10)j = 8i + 7j.
Question 2. A is the point (1, −2) and B is (7, 6). Find AB and its magnitude.
Answer
AB = OB − OA = (7 − 1)i + (6 − (−2))j = 6i + 8j. The magnitude is √(36 + 64) = √100 = 10.
Question 3. Find the unit vector in the direction of 5i − 12j.
Answer
The magnitude is √(25 + 144) = 13. The unit vector is (5i − 12j) ÷ 13 = (5/13)i − (12/13)j.
Check: (5/13)² + (12/13)² = 25/169 + 144/169 = 1.
Question 4. P is (−3, 4) and Q is (5, −2). Find the position vector of the midpoint M of PQ.
Answer
OM = ½(OP + OQ) = ½[(−3i + 4j) + (5i − 2j)] = ½(2i + 2j) = i + j.
Question 5. Find the magnitude of v = −4i − 4j and its direction measured anticlockwise from the positive x-axis.
Answer
|v| = √(16 + 16) = √32 = 4√2 ≈ 5.66.
The reference angle is tan⁻¹(4 ÷ 4) = 45°. Both components are negative, so the vector lies in quadrant 3: θ = 180° + 45° = 225°.
Question 6. Write, in i, j form, a vector of magnitude 20 that makes an angle of 60° with the positive x-axis.
Answer
x = 20 cos 60° = 10 and y = 20 sin 60° = 10√3 ≈ 17.32. The vector is 10i + 17.3j.
Check: √(100 + 300) = √400 = 20.
Question 7. The vector p = 2i + hj has magnitude √29. Find the possible values of h.
Answer
√(4 + h²) = √29, so 4 + h² = 29 and h² = 25.
So h = 5 or h = −5. Both give a vector of the same magnitude, one above and one below the x-axis.
Question 8. Given OA = 2i + 3j and OB = 6i − j, find the unit vector in the direction of AB.
Answer
AB = OB − OA = (6 − 2)i + (−1 − 3)j = 4i − 4j. Its magnitude is √(16 + 16) = 4√2.
The unit vector is (4i − 4j) ÷ (4√2) = (i − j) ÷ √2, which is (√2 ÷ 2)(i − j).
If you got these wrong
- Sign slips in Questions 1, 2, 4 and 8: revisit vectors in component and unit-vector form.
- Angle, length or unit-vector slips in Questions 3 and 5 to 7: read finding magnitude and direction.
Log each slip in the mistake log, and try a timed practice session when you are ready. If you want a teacher to go through your working, see online one-to-one Additional Mathematics tuition.