Worksheets · Precalculus

3D vectors worksheet

Everything you did in the plane carries over with a third component along for the ride. The magnitude picks up one more square under the root, and nothing else changes.

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Pick more than one for a sheet that mixes them.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Vectors in Three Dimensions

Date Period

Find each answer.

  1. Find the magnitude: ⟨10,−2,11⟩
  2. Write PQ→ in component form: P(−5,0,−6), Q(−17,−12,−7)
  3. Find 3u⃗−3v⃗: u⃗=⟨−2,5,−4⟩, v⃗=⟨−6,−9,−1⟩
  4. Find the magnitude: ⟨−2,5,14⟩
  5. Write PQ→ in component form: P(1,−6,−4), Q(−8,−8,2)
  6. Find 3u⃗+3v⃗: u⃗=⟨−9,2,7⟩, v⃗=⟨−5,9,4⟩
  7. Find the magnitude: ⟨8,−9,−12⟩
  8. Write PQ→ in component form: P(3,7,−6), Q(11,11,−25)
  9. Find 3u⃗−3v⃗: u⃗=⟨9,−6,−2⟩, v⃗=⟨−7,−2,−3⟩
  10. Find the magnitude: ⟨−15,10,−6⟩
  11. Write PQ→ in component form: P(−6,−3,3), Q(−7,−7,−5)
  12. Find 2u⃗+3v⃗: u⃗=⟨3,−7,0⟩, v⃗=⟨0,−5,6⟩
  13. Find the magnitude: ⟨4,−8,−1⟩
  14. Write PQ→ in component form: P(−1,1,−3), Q(−5,−7,−22)
  15. Find 2u⃗−3v⃗: u⃗=⟨7,−4,8⟩, v⃗=⟨7,3,3⟩
  16. Find the magnitude: ⟨−18,1,−6⟩
  17. Write PQ→ in component form: P(−2,−4,1), Q(4,−13,3)
  18. Find 2u⃗+3v⃗: u⃗=⟨5,6,−9⟩, v⃗=⟨−6,3,3⟩
  19. Find the magnitude: ⟨4,12,3⟩
  20. Write PQ→ in component form: P(1,6,−6), Q(−2,12,−8)

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

Sigma Prep · sigmaprep.io/worksheets

Name

Vectors in Three DimensionsAnswer keyVersion 1

Date Period

Find each answer.

  1. Find the magnitude: ⟨10,−2,11⟩15
  2. Write PQ→ in component form: P(−5,0,−6), Q(−17,−12,−7)⟨−12,−12,−1⟩
  3. Find 3u⃗−3v⃗: u⃗=⟨−2,5,−4⟩, v⃗=⟨−6,−9,−1⟩⟨12,42,−9⟩
  4. Find the magnitude: ⟨−2,5,14⟩15
  5. Write PQ→ in component form: P(1,−6,−4), Q(−8,−8,2)⟨−9,−2,6⟩
  6. Find 3u⃗+3v⃗: u⃗=⟨−9,2,7⟩, v⃗=⟨−5,9,4⟩⟨−42,33,33⟩
  7. Find the magnitude: ⟨8,−9,−12⟩17
  8. Write PQ→ in component form: P(3,7,−6), Q(11,11,−25)⟨8,4,−19⟩
  9. Find 3u⃗−3v⃗: u⃗=⟨9,−6,−2⟩, v⃗=⟨−7,−2,−3⟩⟨48,−12,3⟩
  10. Find the magnitude: ⟨−15,10,−6⟩19
  11. Write PQ→ in component form: P(−6,−3,3), Q(−7,−7,−5)⟨−1,−4,−8⟩
  12. Find 2u⃗+3v⃗: u⃗=⟨3,−7,0⟩, v⃗=⟨0,−5,6⟩⟨6,−29,18⟩
  13. Find the magnitude: ⟨4,−8,−1⟩9
  14. Write PQ→ in component form: P(−1,1,−3), Q(−5,−7,−22)⟨−4,−8,−19⟩
  15. Find 2u⃗−3v⃗: u⃗=⟨7,−4,8⟩, v⃗=⟨7,3,3⟩⟨−7,−17,7⟩
  16. Find the magnitude: ⟨−18,1,−6⟩19
  17. Write PQ→ in component form: P(−2,−4,1), Q(4,−13,3)⟨6,−9,2⟩
  18. Find 2u⃗+3v⃗: u⃗=⟨5,6,−9⟩, v⃗=⟨−6,3,3⟩⟨−8,21,−9⟩
  19. Find the magnitude: ⟨4,12,3⟩13
  20. Write PQ→ in component form: P(1,6,−6), Q(−2,12,−8)⟨−3,6,−2⟩

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

How to do these

Find the magnitude: ⟨2,−3,6⟩

  1. Square each component: 4, 9 and 36.
  2. They add to 49.
  3. The square root of 49 is 7.

The answer is 7.

Where students go wrong

Leaving the third component out of the magnitude. All three get squared, and a vector with a large z can be much longer than its shadow in the xy-plane looks.

Also called 3D vectors, vectors in space, space vectors or three dimensional vectors.

What you can put on this worksheet

The magnitude of a vector in space
Find the magnitude: ⟨−4,−4,7⟩ → 9
Component form from two points
Write PQ→ in component form: P(−2,4,−1), Q(−6,0,6) → ⟨−4,−4,7⟩
Sums, differences and scalar multiples
Find u⃗+v⃗: u⃗=⟨0,1,3⟩, v⃗=⟨−5,0,1⟩ → ⟨−5,1,4⟩
Unit vectors
Find the unit vector in the direction of v⃗=⟨−4,−4,7⟩ → ⟨−49,−49,79⟩
The dot product
Find u⃗⋅v⃗: u⃗=⟨0,1,3⟩, v⃗=⟨−5,0,1⟩ → 3

Questions about these worksheets

Yes. Every 3d vectors 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 3d vectors 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 two 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 magnitude of a vector in space, component form from two points, sums, differences and scalar multiples, unit vectors and the dot product. Tick as many as you want and set how many of each, or let it spread them evenly.

Leaving the third component out of the magnitude. All three get squared, and a vector with a large z can be much longer than its shadow in the xy-plane looks.