A storage box is 4 metres long, 3 metres wide and 12 metres tall…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A storage box is metres long, metres wide and metres tall. Place one corner at the origin, with the length along the -axis, the width along the -axis and the height along the -axis. A body diagonal joins the origin to the opposite corner . A second body diagonal joins to . A face diagonal runs across the floor from the origin to . A technician must fit a sensor at the point where a body diagonal meets the floor diagonal, and she needs the angle between them to cut the bracket. She also wants the angle between the two body diagonals. Find the length of a body diagonal, the acute angle between the first body diagonal and the floor diagonal, and the acute angle between the two body diagonals.
Need a reminder? The key ideaShow
The acute angle between lines with directions and has . The lines are perpendicular when the dot product is .
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 length. metres.
Step 2. The floor diagonal. Its direction is and its length is metres.
Step 3. The first angle. The dot product with is . So and .
Step 4. The second body diagonal. Its direction is . The dot product with is .
Step 5. The second angle. , so .
Answer: metres, and .
Check. The two body diagonals cross at the centre of the box, so they are not far apart in direction when the box is tall.
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