MediumMathematicsHigh School (11–12)Vectors and Three-Dimensional GeometryNumerical

A cable runs from the top of one mast at A(1, 2, 3) to the top of another mast at B(7, 5, 12), with…

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

A cable runs from the top of one mast at $A(1, 2, 3)$ to the top of another mast at $B(7, 5, 12)$ , with all lengths in metres. A hanger is fixed at the point $P$ so that $AP : PB = 1 : 2$ . A guy wire continues the line of the cable beyond $B$ to a point $Q$ with $AQ : QB = 3 : 1$ . Find the coordinates of $P$ . Then find the $z$ -coordinate of $Q$ .

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

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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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