HardMathematicsHigh School (11–12)Vectors and Three-Dimensional Geometry › Introduction to Three-Dimensional GeometryNumerical
A ski lift line is straight and joins a top station A(2, -3, 4) to a bottom station B(-1, 5, -3)…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A ski lift line is straight and joins a top station to a bottom station . The distances are in metres from a survey point at the origin. The -plane is the level of a lake and the -plane is the line of a long wall, so the lift crosses both planes. Use the section formula to find the ratio in which the -plane divides the segment, as a number. Then find the -coordinate and the -coordinate of the crossing point with the -plane, to two decimal places. Last find the ratio in which the -plane divides the segment, as a decimal number to two places.
Need a reminder? The key ideaShow
The point that divides internally in the ratio is , coordinate by coordinate. For an external division use . A coordinate plane such as divides in the ratio .
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 -plane. The ratio is . It is positive, so the division is internal. As a number it is .
Step 2. The crossing point. With and : , and .
Step 3. The -plane. The ratio is , which is positive. As a decimal it is .
Check. at the crossing point, so it lies on the -plane.
Answer: the ratio is , the crossing point is with and , and the -plane divides in the ratio .
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