A triangular plot of land has its corners at A(1, 1, 1), B(3, 3, 2) and C(2, -1, 3), with…

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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 triangular plot of land has its corners at $A(1, 1, 1)$ , $B(3, 3, 2)$ and $C(2, -1, 3)$ , with coordinates in metres in a model of a sloping field. The surveyor needs the area of the plot. She also needs the height of the triangle from the corner $C$ down to the side $AB$ . This is the width of a path she will build.

Need a reminder? The key ideaShow

The area of the parallelogram on $\vec a$ and $\vec b$ is $\lvert\vec a\times\vec b\rvert$ , and the triangle has half of that. Also $\lvert\vec a\times\vec b\rvert^2 = \lvert\vec a\rvert^2\lvert\vec b\rvert^2 - (\vec a\cdot\vec b)^2$ .

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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