In a high-precision manufacturing bay, a robotic arm maps the positions of three docking pins relative to a fixed calibration origin OOO. The pins have the following position vectors:
∙\bullet∙ Pin PPP has position vector 4i−j+2k4\mathbf{i} - \mathbf{j} + 2\mathbf{k}4i−j+2k ∙\bullet∙ Pin QQQ has position vector 6i+3j−2k6\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}6i+3j−2k ∙\bullet∙ Pin RRR has position vector ki+3j+5kk\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}ki+3j+5k
where kkk is a constant.
The guide rail LLL is a straight line passing through the centers of pins PPP and QQQ.
Determine a vector equation for the line LLL.
Given that the displacement vector PR⃗\vec{PR}PR is perpendicular to the guide rail LLL,
find the value of kkk.
Calculate the area of the triangular space PQRPQRPQR formed by the pins, expressing your answer as a surd in its simplest form.
Practise AQA A Level Maths 1.13 J: Vectors with exam-style questions for A Level Maths. 122 questions covering 1.13.1 Vectors in two and three dimensions, 1.13.2 Magnitude and direction of a vector, 1.13.3 Vector arithmetic, 1.13.4 Position vectors, and 1.13.5 Vectors to solve problems, matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
