Question Type 5: Calculating the angle between two planes Exercises

Show that the planes $x + 2y + 3z = 0$ and $2x + 4y + 6z = 5$ are parallel and find the angle between them.

If the angle between the planes $x + y + z = 0$ and $x - y + 2z = 0$ is $\theta$, find $\cos\theta$.

Compute $\tan\theta$ for the acute angle $\theta$ between the planes $4x + 2y - z = 0$ and $2x - y + 3z = 0$.

Find the angle between the plane $3x + y - 4z = 10$ and the plane perpendicular to the vector $(1,2,3)$.

Determine $k$ so that the angle between the planes $3x + 4y + kz = 6$ and $x - 2y + 2z = 3$ is $60^\circ$.

Find the angle between the planes $3x - y + z = 2$ and $x + y - 2z = 4$.

Calculate the obtuse angle between the planes $5x - 2y + z = 8$ and $-2x + 3y - 4z = 1$.

Determine the angle between the plane through $A(1,0,0)$, $B(0,1,0)$, $C(0,0,1)$ and the plane $2x - y + z = 4$.

Find all real $k$ such that the angle between the planes $x + ky - z = 1$ and $2x - 3y + z = 5$ is $45^\circ$.

Calculate the acute angle between the planes $2x + 3y - 6z = 5$ and $x - y + 2z = 4$.

Calculate the acute angle between the planes $x + 2y + 2z = 7$ and $-x + 4y + z = 3$.

Calculate the acute angle between the planes $\frac{x}{1} + \frac{y}{2} + \frac{z}{3} = 1$ and $4x - y + z = 2$.