The lines $l_1$ and $l_2$ have equations \[\mathbf{r} = \begin{pmatrix} 0 \ 1 \ 2 \end{pmatrix

Question

The lines $l_1$ and $l_2$ have equations $$\mathbf{r} = \begin{pmatrix} 0 \ 1 \ 2 \end{pmatrix} + s \begin{pmatrix} 1 \ 0 \ 3 \end{pmatrix} \text{ and}\hspace{0.5em} \mathbf{r} = \begin{pmatrix} -2 \ 3 \ 1 \end{pmatrix} + t \begin{pmatrix} 2 \ b \ 0 \end{pmatrix}$$ respectively, where $b$ is a constant. For the case where the lines intersect, calculate the acute angle between them. For the case when $b = 1$, find the position vectors of the points $P$ on $l_1$ and $Q$ on $l_2$ such that $O$, $P$ and $Q$ are collinear, where $O$ is the origin.

Accepted Answer

(i) $78.3^\circ$ (ii) $\overrightarrow{OP} = \frac{1}{25}(-14\mathbf{i} + 25\mathbf{j} + 8\mathbf{k}), \overrightarrow{OQ} = \frac{1}{8}(-14\mathbf{i} + 25\mathbf{j} + 8\mathbf{k})$

N2005 P1 Q14

N2005 P1 Q14

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