Two planes $p_1$ and $p_2$ have respective cartesian equations given by \[4x + y + z = 8 \text{

Question

Two planes $p_1$ and $p_2$ have respective cartesian equations given by $$4x + y + z = 8 \text{ and}\hspace{0.5em} 4x + 3y - z = 0.$$ Find the sine of the acute angle between $p_1$ and $p_2$ in the form $\frac{m}{\sqrt{n}}$ where $m$ and $n$ are positive integers to be determined.[3] Verify that the point $A$ with coordinates $(1,0,4)$ lies on $p_1$ and $p_2$. Hence, without the use of a calculator, find the vector equation of the line $l$ formed by the intersection of $p_1$ and $p_2$.[3] It is given that $B$ is a point on $p_1$ with coordinates $(1,3,1)$. Show that $AB$ is perpendicular to $l$ and hence use (a) to deduce exactly the shortest distance from $B$ to $p_2$.[3] Find the cartesian equation of the plane $p_3$ which contains $B$ and is perpendicular to both $p_1$ and $p_2$.[2]

Accepted Answer

(a) $\sin\theta=\dfrac2{\sqrt{13}}$ (b) $\mathbf r=\begin{pmatrix}1\0\4\end{pmatrix}+\lambda\begin{pmatrix}-1\2\2\end{pmatrix}$, $\lambda\in\mathbb R$ (c) $\dfrac{6\sqrt{26}}{13}$ (d) $-x+2y+2z=7$

2025 NYJC P2 Q5

2025 NYJC P2 Q5

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