# Exam-Style Question on Vectors

## A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 399. This question is similar to one that appeared on an A-Level paper. The use of a calculator is allowed.

The points A and B have coordinates $$(3,-2,1)$$ and $$(4, 0, -1)$$ respectively.

The line $$l$$ has the following equation equation:

$$r= \begin{pmatrix} 3 \\ 2 \\ 0 \\ \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ -2 \\ 0 \\ \end{pmatrix}$$

The point C lies on $$l$$ where $$\lambda = 3$$.

(a) Find the coordinates of C.

(b) Find the acute angle ABC, giving your answer to the nearest tenth of a degree.

(c) The point D lies on a line through A and B such that angle ADC is a right angle. Find the coordinates of D.

(d) The point E completes the parallelogram ACBE. Find the coordinates of E.

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