Shape and Space 8

4 marks · Hard

Given the vectors , , and , express the vector as a column vector. Then, calculate its magnitude, giving your answer correct to 1 decimal place.

Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.

Show answer

, \; \text{magnitude} \approx 8.6\) (±0.1)

Must show: Accurate expansion and simplification of vector expression, Correct column vector, Correct magnitude (to 1 d.p.)

Common mistakes: Arithmetic errors in expanding vector expression; Incorrect formula for magnitude (e.g. wrong signs or components)

Show mark scheme and worked solution

Marking points

M1 · 1 mark · Expands each term correctly (multiply each vector by its scalar).

M2 · 1 mark · Adds and subtracts vectors correctly to obtain a single column vector. (follow-through)

A1 · 1 mark · Writes the result as . (follow-through)

A2 · 1 mark · Correctly finds and rounds the magnitude to 1 decimal place (8.6). (follow-through)

Worked solution

Wait, let's check carefully: First: 2p = (2, -4) 3q = (-12, 9) -r = (-2, -1) So, sum first components: 2 + (-12) + (-2) = -12 Second components: -4 + 9 + (-1) = 4 So final vector is (-12, 4).

But in the answer above, original calculation aimed for (-7,5). Should check arithmetic, perhaps one sign was mistaken.

Let’s do the calculation step-by-step: