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 the angle between their normal vectors. The normal vectors are $ \langle 1, 1, -1 \rangle $ and $ \langle 10, -15, 20 \rangle $. The dot product of these vectors is $ 1 \cdot 10 + 1 \cdot (-15) + (-1) \cdot 20 = -15 $. The magnitude of the first vector is $ \sqrt{1^2 + 1^2 + (-1)^2} = \sqrt{3} $, and the magnitude of the second vector is $ \sqrt{10^2 + (-15)^2 + 20^2} = 5 \sqrt{13} $. Therefore, the cosine of the angle between the planes is $ \frac{-15}{\sqrt{3} \cdot 5 \sqrt{13}} = -\frac{1}{\sqrt{39}} $. The angle between the planes is therefore $ \cos^{-1}\left(-\frac{1}{\sqrt{39}}\right) \approx 122^\circ $.

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