<p>The plane ${{p}_{1}}$ contains points $A$ and $B$, with position vectors $\left( \begin{matrix}2 \\0 \\1 \\\end{matrix} \right)$ and $\left( \begin{matrix}8 \\-6 \\3 \\\end{matrix} \right)$ respectively, and is parallel to the line with direction vector $\left( \begin{matrix}7 \\-5 \\3 \\\end{matrix} \right)$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find a cartesian equation of the plane ${{p}_{1}}$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>The plane ${{p}_{2}}$ has equation $\mathbf{r}\cdot (\mathbf{i}-2\mathbf{j}+\mathbf{k})=3$. The planes ${{p}_{1}}$ and ${{p}_{2}}$ meet in the line $l$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find a vector equation of the line $l$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the acute angle between the planes ${{p}_{1}}$ and ${{p}_{2}}$.</div><p style="text-align: right;">[2]</p></li></ol>
<p>It is given that the point $R$ with position vector $-17\mathbf{i}-5\mathbf{j}+10\mathbf{k}$ lies on ${{p}_{2}}$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Find the foot of perpendicular from the point $R$ to the plane ${{p}_{1}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence, find a vector equation of the plane ${{p}_{3}}$ which is a reflection of the plane ${{p}_{2}}$ in the plane ${{p}_{1}}$ in the form $\mathbf{r}\cdot \mathbf{n}=d$, where $d$ is a constant to be determined.</div><p style="text-align: right;">[4]</p></li></ol>

The plane ${{p}_{1}}$ contains points $A$ and $B$, with position vectors $\left( \begin{matrix}2 \\0 \\1 \\\end{matrix} \right)$ and $\left( \begin{matrix}8 \\-6 \\3 \\\end{matrix} \right)$ respectively, and is parallel to the line with direction vector $\left( \begin{matrix}7 \\-5 \\3 \\\end{matrix} \right)$.

Find a cartesian equation of the plane ${{p}_{1}}$.

The plane ${{p}_{2}}$ has equation $\mathbf{r}\cdot (\mathbf{i}-2\mathbf{j}+\mathbf{k})=3$. The planes ${{p}_{1}}$ and ${{p}_{2}}$ meet in the line $l$.

Find the acute angle between the planes ${{p}_{1}}$ and ${{p}_{2}}$.

It is given that the point $R$ with position vector $-17\mathbf{i}-5\mathbf{j}+10\mathbf{k}$ lies on ${{p}_{2}}$.

Find the foot of perpendicular from the point $R$ to the plane ${{p}_{1}}$.

Hence, find a vector equation of the plane ${{p}_{3}}$ which is a reflection of the plane ${{p}_{2}}$ in the plane ${{p}_{1}}$ in the form $\mathbf{r}\cdot \mathbf{n}=d$, where $d$ is a constant to be determined.

2017 CJC J1 MYE Q8

2017 CJC J1 MYE Q8

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