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

Two flat panels lie in the planes x - y + 2z = 3 and 3x + y - z = 2, with lengths in metres…

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

Two flat panels lie in the planes $x - y + 2z = 3$ and $3x + y - z = 2$ , with lengths in metres. They meet along a straight edge. A third panel must pass through this edge and through the point $(1, 1, 1)$ . Write the plane of the third panel as $(x - y + 2z - 3) + \lambda(3x + y - z - 2) = 0$ . Find $\lambda$ . Then write the plane in the form $4x + my + nz = k$ , and find $n$ , $k$ and the distance of the plane from the origin.

Need a reminder? The key ideaShow

Every plane through the line where $P_1 = 0$ and $P_2 = 0$ meet is $P_1 + \lambda P_2 = 0$ . A plane through a line and a point $P$ has normal $\overrightarrow{AP} \times \vec{b}$ . A plane through two parallel lines has normal $(\vec{a}_2 - \vec{a}_1) \times \vec{b}$ .

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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