Review question

Can we show the direction of $P'Q'$ is independent of $P$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8565

Suggestion

The plane $\pi$ has equation $\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}+ \mu \mathbf{c}$ and the plane $\pi'$ has equation $\mathbf{r} = \mathbf{d}+ \lambda'\mathbf{e}+\mu'\mathbf{f}$, where $\lambda$, $\mu$, $\lambda'$ and $\mu'$ are scalar parameters. It is given that

$\mathbf{a}=\begin{pmatrix}1 \\ 0 \\ -1\end{pmatrix}, \mathbf{b}=\begin{pmatrix}2\\ -1 \\ -1\end{pmatrix}, \mathbf{c} = \begin{pmatrix}-1 \\ 2 \\ 1\end{pmatrix}, \mathbf{d} = \begin{pmatrix}0 \\ 1 \\ 1\end{pmatrix}, \mathbf{e} = \begin{pmatrix}p \\ q \\ r\end{pmatrix}, \mathbf{f} = \begin{pmatrix}q\\ r\\ p\end{pmatrix},$

$\big \vert\mathbf{e}\big \vert=\big \vert\mathbf{f}\big \vert = \sqrt{30}$ and that $p$ is negative. The planes $\pi$ and $\pi'$ are parallel. Find $p$, $q$ and $r$.

If $\pi$ and $\pi'$ are parallel, what must be true about the normal of either plane?

How could we use the vector and scalar products here?

Show that $\pi'$ is the reflection of $\pi$ in the origin.

If $P$ is the point $(x,y,z)$, what are the coordinates of the reflection of $P$ in the origin?

The line $L$ has equation $\dfrac{x+1}{9}=\dfrac{y-1}{-6}=\dfrac{z}{-5}$ and lies in $\pi$. The point $P$ is in $\pi$ with parameters $\lambda = l$ and $\mu = m$, and $P'$ is in $\pi'$ with $\lambda' = l$ and $\mu' = m$. The point $Q'$ is the reflection of $P$ in the origin. Show that when $P$ lies on $L$ the direction of $P'Q'$ is independent of the position of $P$ on $L$.

If $P$ is on $L$, can we find the coordinates of $P$ in terms of a parameter $\alpha$?

Can we now find the coordinates of $P'$ and $Q'$ in terms of $\alpha$?