Find a vector equation of the line passing through the points $A$ and $B$ with position vectors

Question

Find a vector equation of the line passing through the points $A$ and $B$ with position vectors $6\mathbf{j}+\mathbf{k}$ and $4\mathbf{i}-2\mathbf{j}+\mathbf{k}$ respectively.[2] Point $C$ has the position vector $3\mathbf{i}+\alpha \mathbf{j}+2\mathbf{k}$, where $\alpha$ is a positive constant. Given that the length of projection of $AC$ on the line $AB$ is $\sqrt{125}$, find the value of $\alpha$.[3] Hence or otherwise, find the distance from the point $C$ to the line $AB$.[3]

Accepted Answer

(i) $\mathbf r=\begin{pmatrix}0\6\1\end{pmatrix}+\lambda\begin{pmatrix}1\-2\0\end{pmatrix},\ \lambda\in\mathbb R$ (ii) $\alpha=20$ (iii) $9$ units

2014 JJC P1 Q8

2014 JJC P1 Q8

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