2023 JPJC Promo Q7
2023 JPJC Promo Q7
7 marks
Promo
The points \(A\) and \(B\) have coordinates \(\left( 3,\ 1,\ 1 \right)\) and \(\left( 1,\ -1,\ 2 \right)\) respectively.
- Find a vector equation of the line \(AB\).[1]
The point \(C\) has coordinates \(\left( 3,\ -16,\ 3 \right)\).
- Find the position vector of the foot of perpendicular from \(C\) to the line \(AB\).[3]
- Find a vector equation of the line which is a reflection of the line \(AC\) in the line \(AB\).[3]
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Answer:(i) A vector equation of the line \({AB}\) is \({\mathbf{r} = \begin{pmatrix} 3 \\ 1 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix}, \lambda \in \mathbb{R}}\). (ii) \({\therefore \overrightarrow{ON} = \begin{pmatrix} -5 \\ -7 \\ 5 \end{pmatrix}}\).
(iii) Vector equation of the line of reflection is \({\mathbf{r} = \begin{pmatrix} 3 \\ 1 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} -16 \\ 1 \\ 6 \end{pmatrix}, \mu \in \mathbb{R}}\).
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