The diagram below shows a rectangular block with its top sliced off. It is given that the lengths of

Question

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 $p$.

Find the value of $a$.

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

Show that $b=0$.
Find the perpendicular distance from $C$ to the plane $p$.

Accepted Answer

(i) $\pi_{DEFG}: \mathbf{r} \cdot \begin{pmatrix} 1 \\ 2 \\ -6 \end{pmatrix} = -6$ (shown) (ii) $a = 1$ (iii) $b = 0$ (iv) $\frac{6}{\sqrt{41}}$ units or $\frac{6\sqrt{41}}{41}$ units

NJC Vectors II Tutorial Q10

NJC Vectors II Tutorial Q10

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