The angle between the planes $2x-y+z=6$ and $x+y+2z=3$ is equal to:

Question

Accepted Answer

To determine the angle between the two planes given by the equations $2x - y + z = 6$ and $x + y + 2z = 3$, we need to find the angle between their normal vectors. The normal vector of a plane $ax + by + cz = d$ is given by the vector $\langle a, b, c \rangle$.

For the plane $2x - y + z = 6$, the normal vector is $\mathbf{n_1} = \langle 2, -1, 1 \rangle$.

For the plane $x + y + 2z = 3$, the normal vector is $\mathbf{n_2} = \langle 1, 1, 2 \rangle$.

The angle $\theta$ between two vectors $\mathbf{a}$ and $\mathbf{b}$ is given by the formula:

$$
\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|}
$$

First, we calculate the dot product $\mathbf{n_1} \cdot \mathbf{n_2}$:

$$
\mathbf{n_1} \cdot \mathbf{n_2} = (2)(1) + (-1)(1) + (1)(2) = 2 - 1 + 2 = 3
$$

Next, we calculate the magnitudes of $\mathbf{n_1}$ and $\mathbf{n_2}$:

$$
\|\mathbf{n_1}\| = \sqrt{2^2 + (-1)^2 + 1^2} = \sqrt{4 + 1 + 1} = \sqrt{6}
$$

$$
\|\mathbf{n_2}\| = \sqrt{1^2 + 1^2 + 2^2} = \sqrt{1 + 1 + 4} = \sqrt{6}
$$

Now, substitute these into the cosine formula:

$$
\cos \theta = \frac{3}{\sqrt{6} \cdot \sqrt{6}} = \frac{3}{6} = \frac{1}{2}
$$

The angle $\theta$ for which $\cos \theta = \frac{1}{2}$ is $\theta = \pi/3$.

Thus, the angle between the planes is $\pi/3$, which corresponds to Option 1.

**Explanation for Other Options:**

- **Option 2 ($\pi/4$)**: This would imply $\cos \theta = \frac{\sqrt{2}}{2}$, which is not equal to $\frac{1}{2}$. Therefore, this option is incorrect.

- **Option 3 ($\cos^{-1}(\frac{1}{\sqrt{6}})$)**: This suggests a different cosine value, $\frac{1}{\sqrt{6}}$, which does not match the calculated $\frac{1}{2}$. Hence, this option is incorrect.

- **Option 4 ($\cos^{-1}(\frac{2}{5})$)**: This implies $\cos \theta = \frac{2}{5}$, which is not equal to $\frac{1}{2}$. Therefore, this option is incorrect.

In conclusion, the correct answer is indeed Option 1, $\pi/3$, as it matches the calculated angle between the planes.