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

A beam of light travels from a lamp at A(3, -1, 1) to a flat mirror and then to a sensor at B(3…

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

A beam of light travels from a lamp at $A(3, -1, 1)$ to a flat mirror and then to a sensor at $B(3, -1, 6)$ . Lengths are in metres. The mirror lies in the plane $2x - 2y + z = 0$ . A student first asks whether the two points are on the same side of the mirror. The values of $2x - 2y + z$ at $A$ and $B$ show this. The beam takes the shortest path that touches the mirror. Find the values of $2x - 2y + z$ at $A$ and at $B$ , the $x$ -coordinate of the image $A'$ of the lamp, and the length of the shortest path. Compare it with the straight distance $AB$ .

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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