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A ceiling in a hall is tilted and lies in the plane 4x - 4y + 7z = 7, with coordinates in metres…

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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 ceiling in a hall is tilted and lies in the plane $4x - 4y + 7z = 7$ , with coordinates in metres. A lamp hangs at the point $P(2, 3, -1)$ . Find the value of $4x - 4y + 7z - 7$ at $P$ , and the shortest distance from the lamp to the ceiling.

Need a reminder? The key ideaShow

The distance of $(x_1, y_1, z_1)$ from $Ax + By + Cz + D = 0$ is $\dfrac{\lvert Ax_1 + By_1 + Cz_1 + D \rvert}{\sqrt{A^2 + B^2 + C^2}}$ . Two planes meet in a line with direction $\vec{n}_1 \times \vec{n}_2$ .

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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A ceiling in a hall is tilted and lies in the plane 4x - 4y + 7z = 7, with coordinates in metres…

The worked solution

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