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Q.
The distance of the plane through $(2, 3, - 1)$ and at right angles to the vector $3 \hat{i} - 4\hat{j} + 7 \hat{k}$ from the origin is
Introduction to Three Dimensional Geometry
Solution:
The equation of the plane through $(2, 3, - 1)$ , and perpendicular to the vector
$3\hat{i} - 4 \hat{j} + 7 \hat{k}$ is $3 (x - 2) + (- 4) (y - 3) + 7 (z - (- 1)) = 0$
or $3 x - 4 y + 7 z + 13 = 0$
Distance of this plane from the origin
$= \frac{\left|3 \times0-4 \times0 + 7 \times 0+13\right|}{\sqrt{3^{2} +\left(-4\right)^{2} +7^{2}}}$