First slide

Planes in 3D

Question

If the planes $2 x + 3 y - z + 5 = 0$ , $x + 2 y - k z + 7 = 0$ are perpendicular then $k =$

Moderate

Solution

Given planes are perpendicular $a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$
$\Rightarrow (2) (1) + (3) (2) + (- 1) (- k) = 0$
$\Rightarrow 2 + 6 + k = 0$ $\Rightarrow k = - 8$

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