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

A rail passes through the origin with direction ratios (1, 1, 1), so a point of the rail is (t, t…

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A rail passes through the origin with direction ratios $(1, 1, 1)$ , so a point of the rail is $(t, t, t)$ , with lengths in metres. Two panels lie in the planes $2x - y + 2z = 3$ and $2x + 3y + 6z = -1$ . A technician must fix a sensor on the rail at a point that is equally far from the two panels. The distance of $(t, t, t)$ from the first panel is $\lvert t - 1 \rvert$ . The distance from the second is $\dfrac{\lvert 11t + 1 \rvert}{7}$ . There are two such points, one on each side of the first panel. Find the smaller value of $t$ , the larger value of $t$ , and the common distance for the smaller value.

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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