MediumMathematicsHigh School (11–12)Vectors and Three-Dimensional GeometryNumerical

A straight pole stands along the line through the origin with direction ratios (2, 3, 6), with…

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A straight pole stands along the line through the origin with direction ratios $(2, 3, 6)$ , with lengths in metres. A mirror disc is fixed at the point $P(5, -3, 8)$ . Find the coordinates of the foot $F$ of the perpendicular from $P$ to the pole, which is also the midpoint of $PP'$ . Then find the image $P'$ of $P$ in the pole, and the distance $PP'$ .

Need a reminder? The key ideaShow

The image of $P$ in the plane $\vec{r} \cdot \vec{n} = k$ is $P' = \vec{p} - 2\,\dfrac{\vec{p} \cdot \vec{n} - k}{\lvert \vec{n} \rvert^2}\,\vec{n}$ . The image in a line is $2F - P$ . The shortest path $A \to M \to B$ off a mirror, with $A$ and $B$ on one side, is $A'B$ .

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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