The planes ${{\mathrm{p}}_{1}}$ and ${{\mathrm{p}}_{2}}$ have equations \(\mathbf{r}\cdot \left(

Question

The planes ${{\mathrm{p}}_{1}}$ and ${{\mathrm{p}}_{2}}$ have equations $\mathbf{r}\cdot \left( \begin{matrix}2 \-2 \1 \\end{matrix} \right)=1$ and $\mathbf{r}\cdot \left( \begin{matrix}-6 \3 \2 \\end{matrix} \right)=-1$ respectively, and meet in the line $l$. Find the acute angle between ${{\mathrm{p}}_{1}}$ and ${{\mathrm{p}}_{2}}$.[3] Find a vector equation for $l$.[4] The point $A\left( 4,3,c \right)$ is equidistant from the planes ${{\mathrm{p}}_{1}}$ and ${{\mathrm{p}}_{2}}$. Calculate the two possible values of $c$.[6]

Accepted Answer

(i) $40.4^\circ$ (ii) $\mathbf r=\begin{pmatrix}-\frac16\-\frac23\0\end{pmatrix}+\lambda\begin{pmatrix}7\10\6\end{pmatrix},\ \lambda\in\mathbb R$ (iii) $c=-49$ or $c=\frac{35}{13}$

N2013 P2 Q4

N2013 P2 Q4

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