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

A glass panel for a roof lies in a plane that meets the three coordinate axes at the points A, B…

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A glass panel for a roof lies in a plane that meets the three coordinate axes at the points $A$ , $B$ and $C$ . Coordinates are in metres. The panel is the triangle $ABC$ , and its centroid is at $(1, 2, 3)$ . The designer knows only the centroid, and the glazier needs the full details. The centroid of a triangle is the mean of its three vertices, so the intercepts are three times the centroid coordinates. Find the intercept on the $z$ -axis. Find the equation of the plane in the form $6x + 3y + 2z = k$ . Then find the distance of the plane from the origin, and the area of the triangle $ABC$ in square metres.

Need a reminder? The key ideaShow

A plane with normal $\vec{n}$ through the point $\vec{a}$ is $\vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n}$ . Through three points, a normal is $\overrightarrow{PQ} \times \overrightarrow{PR}$ . The distance of $Ax + By + Cz = k$ from the origin is $\dfrac{\lvert k \rvert}{\sqrt{A^2 + B^2 + C^2}}$ .

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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