Science:Math Exam Resources/Courses/MATH105/April 2014/Question 01 (a)/Solution 1

The planes are orthogonal if the scalar product of their normal vector is zero.

$\begin{aligned} n_{Q} & = (- 1, 5, - 3) \\ n_{P} & = (3, - \frac{9}{5}, - 4) \end{aligned}$

$\begin{aligned} n_{Q} \cdot n_{P} & = (- 1) \cdot 3 + 5 \cdot (- \frac{9}{5}) + (- 3) \cdot (- 4) = 0 . \end{aligned}$

Hence the two planes are orthogonal.