Exercises for Question Type 5: Calculating the angle between two planes

Question 1

Skill question

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

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Question 2

Skill question

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

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Question 3

Skill question

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

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Question 4

Skill question

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

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Question 5

Skill question

Determine $k$ so that the angle between the planes $3 x + 4 y + k z = 6$ and $x - 2 y + 2 z = 3$ is $6 0^{\circ}$ .

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Question 6

Skill question

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

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Question 7

Skill question

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

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Question 8

Skill question

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

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Question 9

Skill question

Find all real $k$ such that the angle between the planes $x + k y - z = 1$ and $2 x - 3 y + z = 5$ is $4 5^{\circ}$ .

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Question 10

Skill question

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

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Question 11

Skill question

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

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Question 12

Skill question

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

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IB Mathematics Analysis and Approaches Question Type 5: Calculating the Angle Between Two Planes Exercises