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

Question

Quizplus · Accepted Answer

The angle between two planes is given by the formula $\cos \theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2} \cdot \sqrt{a_2^2 + b_2^2 + c_2^2}}$. For the given planes, the normal vectors are $\langle 1, 1, -1 \rangle$ and $\langle 10, -15, 20 \rangle$. Calculating the dot product and magnitudes, we find $\cos \theta = -\frac{1}{3}$, leading to $\theta \approx 122^\circ$.

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