The plane ${{P}_{1}}$ has the equation $-6x-4y+2z=4$. Find the vector equations of the planes such t

Question

The plane ${{P}_{1}}$ has the equation $-6x-4y+2z=4$. Find the vector equations of the planes such that the perpendicular distance from each plane to ${{P}_{1}}$ is $10$ units. [2] The plane ${{P}_{2}}$ has the equation $-x-y+2z=k$, where $k$ is a constant. Find the angle between ${{P}_{1}}$ and ${{P}_{2}}$. [2] The planes ${{P}_{1}}$ and ${{P}_{2}}$ intersect in the line $L$. Show that a possible vector equation of $L$ is $\mathbf{r}=\left( \begin{matrix} 2k-2 \ 2-3k \ 0 \ \end{matrix} ight)+\lambda \left( \begin{matrix} -3 \ 5 \ 1 \ \end{matrix} ight)$, $\lambda \in \mathbb{R}$. [3] The plane ${{P}_{3}}$ has the equation $5x+\beta y+5z=\mu $, where $\beta $, $\mu \in \mathbb{R}$. Given that the line $L$ is contained in the plane ${{P}_{3}}$, find $\beta $ and $\mu $, giving your answer in terms of $k$ if necessary. [2] Given instead that the line $L$ does not intersect ${{P}_{3}}$, what can be said about the values of $\beta $ and $\mu $? [1]

Accepted Answer

(i) $\mathbf{r} \cdot \begin{pmatrix} -3 \\ -2 \\ 1 \end{pmatrix} = 2 \pm 10\sqrt{14}$ (ii) $40.2°$ (iii) $L: \mathbf{r} = \begin{pmatrix} 2k-2 \\ 2-3k \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} -3 \\ 5 \\ 1 \end{pmatrix}$ (iv) $\mu = 4k - 6$ (v) $\beta = 2$, $\mu \ eq 4k - 6$

2022 ACJC Promo Q10

2022 ACJC Promo Q10

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