Vectors: dot & cross product worksheet with answers

Free printable Vectors: dot & cross product worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
xyza

Find the y-component ûy of the unit vector û that points in the direction of a = (−3, 4, 3).

a = (−3, 4, 3)
ûy =
2.Easy
xyza

Find the magnitude |a| of the vector a = (9, 1, −8).

a = (9, 1, −8)
|a| =
3.Medium
xyzab

Find the angle θ between the vectors a = (−1, −1, 3) and b = (5, 6, 4), in degrees.

a = (−1, −1, 3)b = (5, 6, 4)
θ = °
4.Medium
xyzab

Find the dot product a·b of a = (5, 7, −2) and b = (0, −6, −3).

a = (5, 7, −2)b = (0, −6, −3)
a·b =
5.Hard
xyzrFOA

A force F = (−20, −60, 20) N acts at point A(2, 2, 1). Coordinates are in metres. Find the z-component Mz of the moment of the force about the origin O(0, 0, 0).

F = (−20, −60, 20) NA(2, 2, 1) mO(0, 0, 0) m
Mz = N·m
6.Hard
xyzab

The unit vector n̂ is perpendicular to both a = (−2, 1, −5) and b = (5, −5, 1) and has a positive z-component. Find its x-component n̂x.

a = (−2, 1, −5)b = (5, −5, 1)n̂z > 0
n̂x =

Answer key

  1. 1. ûy = 0.686
  2. 2. |a| = 12.08
  3. 3. θ = 88.03 °
  4. 4. a·b = −36
  5. 5. Mz = −80 N·m
  6. 6. n̂x = −0.714

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