<p>The diagram below shows a rectangular block with its top sliced off. It is given that the lengths of $OA$, $OC$, $OD$, and $OH$ are $6$cm, $3$cm, $1$cm and $3$cm respectively. It is further given that the quadrilateral $ACGE$ is a rectangle.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1761312209271-xe0mzz0jij.png" width="400" class="aligncenter" /></figure>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the equation of the plane $DEFG$ is given by</div><div class="list-item-paragraph" style="text-align: center">$\mathbf{r}=\mathbf{k}+\lambda \left( 6\mathbf{i}+3\mathbf{j}+2\mathbf{k} \right)+\mu \left( 2\mathbf{i}-\mathbf{j} \right)$,</div><div class="list-item-paragraph">where $\lambda $ and $\mu $ are real parameters. Hence find the equation of the plane $DEFG$ in the form $\mathbf{r}\cdot \mathbf{n}=d$.</div></li></ol>
<p>The plane $p$ has equation $ax+2y-6z=b$, where $a$ and $b$ are constants. It is given that the line $DF$ does not intersect $p$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the value of $a$.</div></li></ol>
<p>It is further given that the plane $p$ is equidistant between the plane $DEFG$ and the point $C$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Show that $b=0$.</div></li><li><div class="list-item-paragraph">Find the perpendicular distance from $C$ to the plane $p$.</div></li></ol>

The diagram below shows a rectangular block with its top sliced off. It is given that the lengths of $OA$, $OC$, $OD$, and $OH$ are $6$cm, $3$cm, $1$cm and $3$cm respectively. It is further given that the quadrilateral $ACGE$ is a rectangle.

Show that the equation of the plane $DEFG$ is given by

$\mathbf{r}=\mathbf{k}+\lambda \left( 6\mathbf{i}+3\mathbf{j}+2\mathbf{k} \right)+\mu \left( 2\mathbf{i}-\mathbf{j} \right)$,

where $\lambda $ and $\mu $ are real parameters. Hence find the equation of the plane $DEFG$ in the form $\mathbf{r}\cdot \mathbf{n}=d$.

The plane $p$ has equation $ax+2y-6z=b$, where $a$ and $b$ are constants. It is given that the line $DF$ does not intersect $p$.

It is further given that the plane $p$ is equidistant between the plane $DEFG$ and the point $C$.

Find the perpendicular distance from $C$ to the plane $p$.

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International Baccalaureate Higher Level Mathematics

NJC Vectors II Tutorial Q10

NJC Vectors II Tutorial Q10

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