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A robot arm carries a sensor at the point P(2, 3, -6), with distances in centimetres…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A robot arm carries a sensor at the point , with distances in centimetres. For a calibration test a computer replaces the sensor reading by its reflection three times in a row. First it takes the image of in the -axis and calls it . Then it takes the image of in the -plane and calls it . Last it takes the image of in the -plane and calls it . Find the -coordinate of . Then find the distance between and in centimetres, to two decimal places. Check each step with the rule for the signs, and do not round before the last step.
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. Image in the -axis. .
Step 2. Image in the -plane. Change the sign of : .
Step 3. Image in the -plane. Change the sign of : . The -coordinate is .
Step 4. The distance. and differ in by and in by , and they have the same . The distance is .
Check. After three reflections the signs of and are changed and the sign of is back to the start. The result is the image of in the -axis, and is twice the distance of from the -axis.
Answer: , and .
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