MediumMathematicsHigh School (11–12)Vectors and Three-Dimensional Geometry › Introduction to Three-Dimensional GeometryNumerical
A lamp hangs in a room at the point (3, -2, 4), with distances in metres…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A lamp hangs in a room at the point , with distances in metres. A large plane mirror covers the wall that lies in the -plane, so the mirror is the plane . The lamp is on one side of the mirror. A person standing on the other side sees the image of the lamp behind the mirror. Find the -coordinate of the image. Then find the distance between the lamp and its image.
Need a reminder? The key ideaShow
The foot of the perpendicular on a coordinate plane makes one coordinate zero. The image in a plane changes one sign, the image in an axis changes two signs, and the image in the origin changes all three. The foot is the midpoint of the point and its image.
The worked solution
Have you tried it?
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.
Step 1. The image. The image in the -plane is . Its -coordinate is .
Step 2. The distance. The lamp is in front of the mirror. The image is behind it. The distance between them is .
Check. The midpoint of and is , a point on the mirror.
Answer: the image has and is from the lamp.
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