The lines $l_1$ and $l_2$ have equations \[l_1 : \mathbf{r} = \begin{pmatrix} 1 \ 2 \0 \end{pm

Question

The lines $l_1$ and $l_2$ have equations $$l_1 : \mathbf{r} = \begin{pmatrix} 1 \ 2 \0 \end{pmatrix} + \lambda \begin{pmatrix} -2 \ 1 \ 2 \end{pmatrix}, \\quad l_2 : \mathbf{r} = \begin{pmatrix} 0 \ 5 \ -8 \end{pmatrix} + \mu \begin{pmatrix} 2 \ -2 \ 3 \end{pmatrix}$$ where $\lambda$ and $\mu$ are parameters. Show that $l_1$ and $l_2$ are skew lines.[2] The point $P$ has coordinates $(1, 5, 0)$ and the plane $\pi_1$ has equation $y - z - 2 = 0$. Find the coordinates of $P'$, the reflection of $P$ in $\pi_1$.[4] Find the cartesian equation of the plane $\pi_2$ containing $P$ and $l_1$.[3]

Accepted Answer

(i) $l_1$ and $l_2$ are skew (ii) $P'=(1,2,3)$ (iii) $\pi_2:x+z=1$

2025 SAJC P1 Q3

2025 SAJC P1 Q3

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