A site engineer lowers a straight cable into a pit…

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Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A site engineer lowers a straight cable into a pit. The top of the cable is at $A(1, -2, 4)$ above the ground, and the bottom is at $B(6, 8, -6)$ in the pit, with lengths in metres. The ground is the plane $z = 0$ , so the cable crosses the ground between $A$ and $B$ . Find the ratio $AP : PB$ in which the ground divides the cable, and find the point $P$ where the cable crosses it.

Need a reminder? The key ideaShow

If $AP : PB = m : n$ , then $\vec p = \dfrac{n\vec a + m\vec b}{m + n}$ . For an external division use $\dfrac{m\vec b - n\vec a}{m - n}$ . The centroid of a triangle is $\dfrac{\vec a + \vec b + \vec c}{3}$ , and three points are collinear when $\overrightarrow{AC}$ is a number times $\overrightarrow{AB}$ .

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.

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