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

A flat roof panel passes through the point A(2, -1, 4), with coordinates in metres, and a…

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

A flat roof panel passes through the point $A(2, -1, 4)$ , with coordinates in metres, and a vertical-looking support rod, which is the normal of the panel, points along $2\hat{i} + 3\hat{j} + 6\hat{k}$ . The origin is a reference point on the ground. Write the equation of the panel in the form $2x + 3y + 6z = d$ . Find $d$ , the distance of the panel from the origin, and the point where the panel meets the $z$ -axis.

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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