The cartesian equation of the plane ${{\pi }_{1}}$ is given by $4x-5y+z=5$. Find the perpendicul

Question

The cartesian equation of the plane ${{\pi }_{1}}$ is given by $4x-5y+z=5$. Find the perpendicular distance from the origin $O$ to ${{\pi }_{1}}$.[1] Find the acute angle between ${{\pi }_{1}}$ and the $y$-axis.[2] The cartesian equation of the plane ${{\pi }_{2}}$ is given by $2x+5y+3z=5$. Find a vector equation of $l$, the line of intersection of ${{\pi }_{1}}$ and ${{\pi }_{2}}$.[2] The cartesian equation of the plane ${{\pi }_{3}}$ is given by $3x-3y+z=0$. Describe the geometrical relationship between $l$ and ${{\pi }_{3}}$, justifying your answer.[3] Find an equation of the plane $p$ which contains the point $\left( 3,\,-5,\,1 \right)$ and is perpendicular to ${{\pi }_{1}}$, ${{\pi }_{2}}$ and ${{\pi }_{3}}$.[2]

Accepted Answer

(i) $\frac{5}{\sqrt{42}}$ (ii) ${{50.5}^{\circ }}$ ($1$ d.p.) (iii) $l:\mathbf{r}=\left( \begin{matrix} 1 \ 0 \ 1 \ \end{matrix} \right)+\lambda \left( \begin{matrix} 2 \ 1 \ -3 \ \end{matrix} \right)$, where $\lambda \in \mathbb{R}$. (iv) $l$ is parallel to ${{\pi }_{3}}$ and does not lie on ${{\pi }_{3}}$. (v) $\mathbf{r}\cdot \left( \begin{matrix} 2 \ 1 \ -3 \ \end{matrix} \right)=-2$

2022 RI Promo Q9

2022 RI Promo Q9

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