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

A tent frame is a tetrahedron ABCD with corners A(1, 0, 0), B(0, 2, 0), C(0, 0, 3) and D(1, 1, 1)…

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

A tent frame is a tetrahedron $ABCD$ with corners $A(1, 0, 0)$ , $B(0, 2, 0)$ , $C(0, 0, 3)$ and $D(1, 1, 1)$ , with lengths in metres. The canvas of the floor is the triangle $ABC$ . The designer wants the volume of air in the tent, the area of the floor and the height of the top corner $D$ above the plane of the floor. She uses the triple product for the volume, and checks the height with the volume and the area.

Need a reminder? The key ideaShow

The planes bisecting the angle between two planes are $\dfrac{P_1}{\lvert \vec{n}_1 \rvert} = \pm\dfrac{P_2}{\lvert \vec{n}_2 \rvert}$ . The plane through a point meeting two lines is the intersection of the plane through the point and each line. A tetrahedron has volume $\dfrac{1}{6}\lvert \overrightarrow{AB} \cdot (\overrightarrow{AC} \times \overrightarrow{AD}) \rvert$ .

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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