Worksheets · Precalculus

Lines and planes in space worksheet

A plane is fixed by one point on it and a vector perpendicular to it. The components of that normal vector become the coefficients of x, y and z, and the point tells you the constant. A line is given parametrically, so putting a value of t in gives a point on it.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Lines and Planes in Space

Date Period

Find each.

  1. Write the equation of a plane

    Find the equation of the plane through (4,4,−2) with normal vector ⟨3,−4,2⟩
  2. Find the equation of the plane through (−2,−6,6) parallel to 2x−3y−2z=3
  3. Find the equation of the plane through (5,−3,−5) with normal vector ⟨4,−6,5⟩
  4. Find the equation of the plane through (0,−2,3) with normal vector ⟨1,−3,−1⟩

  1. Find a point on a 3D line

    A line is x=2+7t, y=3−t, z=−1+2t. Find the point at t=5
  2. A line is x=−4+t, y=1+4t, z=3+2t. Find the point at t=3
  3. A line is x=−4+2t, y=7+3t, z=−4+t. Find the point at t=7
  4. A line is x=−3+5t, y=6+3t, z=5−4t. Find the point at t=5

  1. Write a line through two points

    Write parametric equations of the line through (−1,−3,−4) and (−1,−8,−5).
  2. Write parametric equations of the line through (−3,2,6) and (2,4,11).
  3. Write parametric equations of the line through (−5,0,0) and (−8,3,−3).
  4. Write parametric equations of the line through (−3,−4,−3) and (2,1,−7).

  1. Write a plane through three points

    Find the equation of the plane through (2,0,0), (0,−1,0), (0,0,−4)
  2. Find the equation of the plane through (3,4,−3), (−4,3,−2), (3,0,4)
  3. Find the equation of the plane through (4,4,3), (−4,−3,−4), (4,−2,2)
  4. Find the equation of the plane through (−1,0,0), (0,−5,0), (0,0,−2)

  1. Find where a line meets a plane

    Where does the line x=3+2t, y=−2+2t, z=−3−2t meet the plane 2x+2y+3z=−13?
  2. Where does the line x=2+t, y=−3+3t, z=2−2t meet the plane −x−y+z=−9?
  3. Find the distance from (−2,3,3) to the plane 2x−y+2z=−13
  4. Where does the line x=−3+2t, y=3−t, z=−2+t meet the plane −x+3z=−2?

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Lines and Planes in SpaceAnswer keyVersion 1

Date Period

Find each.

  1. Write the equation of a plane

    Find the equation of the plane through (4,4,−2) with normal vector ⟨3,−4,2⟩3x−4y+2z=−8
  2. Find the equation of the plane through (−2,−6,6) parallel to 2x−3y−2z=32x−3y−2z=2
  3. Find the equation of the plane through (5,−3,−5) with normal vector ⟨4,−6,5⟩4x−6y+5z=13
  4. Find the equation of the plane through (0,−2,3) with normal vector ⟨1,−3,−1⟩x−3y−z=3

  1. Find a point on a 3D line

    A line is x=2+7t, y=3−t, z=−1+2t. Find the point at t=5(37,−2,9)
  2. A line is x=−4+t, y=1+4t, z=3+2t. Find the point at t=3(−1,13,9)
  3. A line is x=−4+2t, y=7+3t, z=−4+t. Find the point at t=7(10,28,3)
  4. A line is x=−3+5t, y=6+3t, z=5−4t. Find the point at t=5(22,21,−15)

  1. Write a line through two points

    Write parametric equations of the line through (−1,−3,−4) and (−1,−8,−5).Sample answer:
    x=−1, y=−3−5t, z=−4−t
  2. Write parametric equations of the line through (−3,2,6) and (2,4,11).Sample answer:
    x=−3+5t, y=2+2t, z=6+5t
  3. Write parametric equations of the line through (−5,0,0) and (−8,3,−3).Sample answer:
    x=−5−3t, y=3t, z=−3t
  4. Write parametric equations of the line through (−3,−4,−3) and (2,1,−7).Sample answer:
    x=−3+5t, y=−4+5t, z=−3−4t

  1. Write a plane through three points

    Find the equation of the plane through (2,0,0), (0,−1,0), (0,0,−4)2x−4y−z=4
  2. Find the equation of the plane through (3,4,−3), (−4,3,−2), (3,0,4)3x−49y−28z=−103
  3. Find the equation of the plane through (4,4,3), (−4,−3,−4), (4,−2,2)35x+8y−48z=28
  4. Find the equation of the plane through (−1,0,0), (0,−5,0), (0,0,−2)10x+2y+5z=−10

  1. Find where a line meets a plane

    Where does the line x=3+2t, y=−2+2t, z=−3−2t meet the plane 2x+2y+3z=−13?(−3,−8,3)
  2. Where does the line x=2+t, y=−3+3t, z=2−2t meet the plane −x−y+z=−9?(4,3,−2)
  3. Find the distance from (−2,3,3) to the plane 2x−y+2z=−134
  4. Where does the line x=−3+2t, y=3−t, z=−2+t meet the plane −x+3z=−2?(−1,2,−1)

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Find the equation of the plane through (1,2,3) with normal vector ⟨2,−1,4⟩

  1. The normal gives the coefficients: 2x - y + 4z.
  2. Put the point in: 2 - 2 + 12.

The answer is 2x−y+4z=12.

Where students go wrong

Using the point as the coefficients and the normal as the constant. It is the other way round: the normal vector supplies the coefficients.

Also called equation of a plane, parametric equations of a line or normal vector.

What you can put on this worksheet

The equation of a plane
Find the equation of the plane through (−4,0,1) with normal vector ⟨3,1,2⟩ → 3x+y+2z=−10
A point on a line in space
A line is x=t, y=1+t, z=2+t. Find the point at t=2 → (2,3,4)
A line through two points
Write parametric equations of the line through (0,1,3) and (−5,1,4). → Sample answer:
x=−5t, y=1, z=3+t
A plane through three points
Find the equation of the plane through (4,0,0), (0,1,0), (0,0,−4) → x+4y−z=4
Where a line meets a plane, and distance to a plane
Where does the line x=0−2t, y=1, z=2−2t meet the plane 2x+2y+2z=−2? → (−2,1,0)

Questions about these worksheets

Yes. Every lines and planes in space sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the lines and planes in space questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 5: the equation of a plane, a point on a line in space, a line through two points, a plane through three points and where a line meets a plane, and distance to a plane. Tick as many as you want and set how many of each, or let it spread them evenly.

Using the point as the coefficients and the normal as the constant. It is the other way round: the normal vector supplies the coefficients.