<p>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 $\overrightarrow{OR}=5\overrightarrow{OQ}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $R$ is on the line $l$ whose vector equation is</div><div class="list-item-paragraph" style="text-align: center">$\mathbf{r}=-15\mathbf{i}+10\mathbf{j}+5\mathbf{k}+\theta (\mathbf{i}+3\mathbf{j}-4\mathbf{k})$, where $\theta $ is a real parameter.</div></li><li><div class="list-item-paragraph">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 $L$.</div><div class="list-item-paragraph">Hence find the position vector of the reflection of the point $A$ in the plane $p$.</div></li></ol>

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 $p$.

NJC Vectors II Tutorial Q6

NJC Vectors II Tutorial Q6

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