Referring to the origin $O$, two planes $\pi_1$ and $\pi_2$ are given by \[\pi_1 : \mathbf{r}

Question

Referring to the origin $O$, two planes $\pi_1$ and $\pi_2$ are given by $$\pi_1 : \mathbf{r} \cdot \begin{pmatrix} 1 \ 2 \ 2 \end{pmatrix} = 18 \quad \text{and} \quad \pi_2 : \mathbf{r} \cdot \begin{pmatrix} 2 \ -1 \ 2 \end{pmatrix} = 3.$$ Given that point $A(2, 5, 2)$ lies on $\pi_2$, find $\overrightarrow{OB}$, where $B$ is the image of $A$ when reflected in the plane $\pi_1$.[4] Find a vector equation of the line of intersection of $\pi_1$ and $\pi_2$.[1] Find the Cartesian equation of the plane which is the image of $\pi_2$ when $\pi_2$ is reflected in $\pi_1$.[3]

Accepted Answer

(i) $\overrightarrow{OB}=\dfrac19\begin{pmatrix}22\53\26\end{pmatrix}$ (ii) $\mathbf r=\begin{pmatrix}0\5\4\end{pmatrix}+\mu\begin{pmatrix}6\2\-5\end{pmatrix},\ \mu\in\mathbb R$ (iii) $10x-25y+2z=-117$

Vectors - Perpendicular Projection and Reflection in a Plane

Vectors - Perpendicular Projection and Reflection in a Plane

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