EasyMathematicsHigh School (11–12)Vectors and Three-Dimensional Geometry › Three-Dimensional GeometryNumerical
A radio mast stands at the origin of a site plan, with its top at the point A(0, 0, 12)…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A radio mast stands at the origin of a site plan, with its top at the point . A steel guy-wire runs from the top of the mast to an anchor in the ground at the point . All coordinates are in metres, and the -axis points straight up. The engineer wants the length of the wire and the direction in which it runs, taken from the top of the mast to the anchor. She also wants the acute angle between the wire and the mast, because the anchor bolt must be set at that angle.
Need a reminder? The key ideaShow
For a line with direction ratios the direction cosines are , , with , and . The line through two points has direction ratios equal to the differences of the coordinates.
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 direction ratios. .
Step 2. The length. metres.
Step 3. The direction cosines. From the top to the anchor they are , and .
Step 4. The angle with the mast. The angle with the upward -axis is . The wire is a line, so the acute angle is .
Answer: metres, , and an angle of about .
Check. .
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