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<p>The line ${{l}_{1}}$ passes through the point $A\left( 1,\ 2,\ 4 \right)$ and point $B\left( -1,\ -1,\ 3 \right)$. The line ${{l}_{2}}$ has equation</p>
<p style="text-align: center">$\mathbf{r}=\left( \begin{matrix} \begin{matrix} 3 \\ -5 \\ 2 \\\end{matrix} \\\end{matrix} \right)+t\left( \begin{matrix} \begin{matrix} 6 \\ -7 \\ 2 \\\end{matrix} \\\end{matrix} \right)$, $t\in \mathbb{R}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Explain why ${{l}_{1}}$ and ${{l}_{2}}$ are skew lines.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find an equation of the plane, in scalar product form, that includes the midpoint of $AB$ and the line ${{l}_{2}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the coordinates of $C$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>The plane ${{\pi }_{2}}$ has equation $3x-4y+\lambda z=\mu $ and line ${{l}_{1}}$ does not intersect ${{\pi }_{2}}$.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find the value of $\lambda $. Hence find the acute angle between the planes ${{\pi }_{1}}$ and ${{\pi }_{2}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">If the distance between ${{\pi }_{2}}$ and ${{l}_{1}}$ is 2 units, find the exact values of $\mu $.</div><p style="text-align: right;">[3]</p></li></ol>

The line ${{l}_{1}}$ passes through the point $A\left( 1,\ 2,\ 4 \right)$ and point $B\left( -1,\ -1,\ 3 \right)$. The line ${{l}_{2}}$ has equation

$\mathbf{r}=\left( \begin{matrix} \begin{matrix} 3 \\ -5 \\ 2 \\\end{matrix} \\\end{matrix} \right)+t\left( \begin{matrix} \begin{matrix} 6 \\ -7 \\ 2 \\\end{matrix} \\\end{matrix} \right)$, $t\in \mathbb{R}$.

The plane ${{\pi }_{2}}$ has equation $3x-4y+\lambda z=\mu $ and line ${{l}_{1}}$ does not intersect ${{\pi }_{2}}$.

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