Question

The angle between the planes x=√3 and z = √2 is equal to

• A. \(\frac{\pi}{6}\)
• B. \(\frac{\pi}{4}\)
• C. \(\frac{\pi}{3}\)
• D. \(\frac{\pi}{2}\)
• E. 0

Answer 1

The Correct Option is D

The equations of the planes are given by:
\( x = \sqrt{3} \)
\( z = \sqrt{2} \)

These are vertical planes that are parallel to the \( yz \)-plane (for the first plane) and the \( xy \)-plane (for the second plane).

Since one plane is vertical along the \( x \)-axis and the other is vertical along the \( z \)-axis, the angle between these two planes is the angle between the \( x \)-axis and the \( z \)-axis, which is \( \frac{\pi}{2} \).

So, the correct option is (D) : \(\frac{\pi}{2}\)

Answer 2

The equation \(x = \sqrt{3}\) represents a plane parallel to the yz-plane. Its normal vector is \(\bar{n_1} = (1, 0, 0)\).

The equation \(z = \sqrt{2}\) represents a plane parallel to the xy-plane. Its normal vector is \(\bar{n_2} = (0, 0, 1)\).

The angle \(\theta\) between the two planes is the angle between their normal vectors. We use the formula \(\cos\theta = \frac{|\bar{n_1} \cdot \bar{n_2}|}{|\bar{n_1}| |\bar{n_2}|}\).

We have \(\bar{n_1} \cdot \bar{n_2} = (1)(0) + (0)(0) + (0)(1) = 0\).

Also, \(|\bar{n_1}| = \sqrt{1^2 + 0^2 + 0^2} = 1\) and \(|\bar{n_2}| = \sqrt{0^2 + 0^2 + 1^2} = 1\).

Therefore, \(\cos\theta = \frac{|0|}{1 \cdot 1} = 0\), which means \(\theta = \frac{\pi}{2}\).