Practice Planes in 3D in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A flat, infinite surface in 3D space described by ax+by+cz=dax + by + cz = d, where ⟨a,b,c⟩\langle a, b, c \rangle is the normal vector.

Think of a plane as a perfectly flat, infinite floor that can be tilted at any angle in space. A horizontal floor is one plane; tilt it and you get another. To describe which tilt you have, imagine sticking a pole straight up out of the floor—that pole is the normal vector, and it captures the exact orientation of the surface. Any flat sheet in 3D, no matter how it's angled, is completely determined by where it sits and which direction its pole points.

Showing a random 20 of 50 problems.

Example 1

medium
Find the distance from the point (3,−1,2)(3, -1, 2) to the plane 2x+y−2z=42x + y - 2z = 4.

Example 2

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Find the direction vector of the line where x+y+z=6x + y + z = 6 and 2x−y+z=32x - y + z = 3 intersect.

Example 3

easy
Find the zz-intercept of the plane 2x+3y+4z=122x + 3y + 4z = 12.

Example 4

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Find the distance from the point (1,1,1)(1, 1, 1) to the plane 2x−2y+z=52x - 2y + z = 5.

Example 5

easy
Are the planes x−2y+z=4x - 2y + z = 4 and −2x+4y−2z=1-2x + 4y - 2z = 1 parallel?

Example 6

easy
Write the equation of the plane with normal vector ⟨2,−3,1⟩\langle 2, -3, 1 \rangle passing through the point (1,4,−2)(1, 4, -2).

Example 7

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Find a plane through A(1,0,2)A(1,0,2), B(2,1,0)B(2,1,0), C(0,−1,1)C(0,-1,1).

Example 8

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Is ⟨1,2,−3⟩\langle 1, 2, -3 \rangle parallel to the plane x+y+z=0x + y + z = 0?

Example 9

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Is the line x=1+t, y=2, z=3−tx=1+t,\ y=2,\ z=3-t parallel to the plane x+z=5x+z=5?

Example 10

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Find the equation of the plane perpendicular to the line r⃗(t)=(1,2,3)+t⟨4,−1,2⟩\vec{r}(t)=(1,2,3)+t\langle 4,-1,2\rangle passing through (0,1,0)(0,1,0).

Example 11

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Find the foot of the perpendicular from (2,3,4)(2, 3, 4) to the plane x+y+z=0x + y + z = 0.

Example 12

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Find dd so that the plane 2x−y+2z=d2x - y + 2z = d is tangent to the sphere x2+y2+z2=9x^2 + y^2 + z^2 = 9 (i.e., distance 3 from origin).

Example 13

challenge
Find the reflection of the point (1,2,3)(1, 2, 3) across the plane x+y+z=0x + y + z = 0.

Example 14

easy
Is ⟨1,1,0⟩\langle 1,1,0 \rangle parallel to the plane x+y+z=5x + y + z = 5 (i.e., perpendicular to its normal)?

Example 15

easy
The plane z=4z = 4 has what normal vector?

Example 16

hard
Find the distance from (1,2,0)(1, 2, 0) to the plane through (0,0,0)(0,0,0), (1,1,0)(1,1,0), (0,1,1)(0,1,1).

Example 17

easy
Does the point (1,2,3)(1, 2, 3) lie on the plane 2x−y+z=32x - y + z = 3?

Example 18

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Where does the line x=t, y=t, z=tx=t,\ y=t,\ z=t meet the plane x+y+z=6x+y+z=6?

Example 19

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Find the line of intersection direction of planes x+y+z=1x+y+z=1 and x−y+z=3x-y+z=3.

Example 20

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Find the xx-intercept of the plane 3x+6y+2z=123x + 6y + 2z = 12.
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Related Concepts

Lines in 3D