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

A mirror plane must send the point A(1, 2, 3) to the point B(5, -2, 5), with lengths in metres…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A mirror plane must send the point $A(1, 2, 3)$ to the point $B(5, -2, 5)$ , with lengths in metres. The mirror is the perpendicular bisector plane of $AB$ . Find the length $AB$ . Write the plane in the form $2x - 2y + z = k$ and find $k$ . Then find the distance of the mirror from the origin.

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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