Find, correct to the nearest degree, the angle between the planes. $$x + y - z = 1,10 x - 15 y + 20 z = 5$$
A) $122 ^ { \circ }$
B) $12 ^ { \circ }$
C) $22 ^ { \circ }$
D) $102 ^ { \circ }$
E) none of these

Question

Quizplus · Accepted Answer

The angle between the planes is given by the formula:$$\cos 	heta = \frac{|n_1 \cdot n_2|}{\Vert n_1 \Vert \Vert n_2 \Vert}$$where $n_1$ and $n_2$ are the normal vectors to the planes.For the first plane, the normal vector is $(1, 1, -1)$.For the second plane, the normal vector is $(10, -15, 20)$.So,$$\cos 	heta = \frac{|1 \cdot 10 + 1 \cdot (-15) + (-1) \cdot 20|}{\sqrt{1^2 + 1^2 + (-1)^2} \sqrt{10^2 + (-15)^2 + 20^2}} = \frac{-25}{\sqrt{3} \sqrt{625}} = -\frac{1}{5}$$Therefore, $	heta = \cos^{-1}\left(-\frac{1}{5}ight) \approx 122^\circ$.

Find, Correct to the Nearest Degree, the Angle Between the Planes