With respect to the origin, the point $A$ has position vector \(\mathbf{i}+3\mathbf{j}-4\mathbf{k}

Question

With respect to the origin, the point $A$ has position vector $\mathbf{i}+3\mathbf{j}-4\mathbf{k}$, and the point $B$ has position vector $-12\mathbf{i}+8\mathbf{j}+4\mathbf{k}$. The point $P$ lies on the line $OA$. The points $Q$ and $R$ are such that $\overrightarrow{PQ}=\frac{1}{3}\overrightarrow{QB}$ and $\overrightarrow{OR}=5\overrightarrow{OQ}$. Show that $R$ is on the line $l$ whose vector equation is $\mathbf{r}=-15\mathbf{i}+10\mathbf{j}+5\mathbf{k}+\theta (\mathbf{i}+3\mathbf{j}-4\mathbf{k})$, where $\theta $ is a real parameter. The plane $p$ contains the line $l$ and is parallel to the vector $11\mathbf{i}+7\mathbf{j}+8\mathbf{k}$. The line through $A$ perpendicular to $p$ meets $p$ at $L$. Find the position vector of $L$. Hence find the position vector of the reflection of the point $A$ in the plane $p$.

Accepted Answer

(i) $\overrightarrow{OR}=\begin{pmatrix}-15\10\5\end{pmatrix}+\theta\begin{pmatrix}1\3\-4\end{pmatrix}$ (ii) $\overrightarrow{OL}=\dfrac19\begin{pmatrix}-101\137\19\end{pmatrix}$, reflection $\dfrac19\begin{pmatrix}-211\247\74\end{pmatrix}$

NJC Vectors II Tutorial Q6

NJC Vectors II Tutorial Q6

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