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 z-component ûz of the unit vector û that points in the direction of a = (−1, 6, −4).

a = (−1, 6, −4)
ûz =
2.Easy
xy60°F = 30 N

A force of 30 N acts in the xy-plane at 60° measured anticlockwise from the positive x-axis. Find its x-component Fx.

F = 30 Nθ = 60°
Fx = N
3.Medium
xyzab

Find the angle θ between the vectors a = (0, 2, −3) and b = (2, 2, 2), in degrees.

a = (0, 2, −3)b = (2, 2, 2)
θ = °
4.Medium
xyzab

Find the angle θ between the vectors a = (2, −1, 0) and b = (2, 5, −6), in degrees.

a = (2, −1, 0)b = (2, 5, −6)
θ = °
5.Hard
xyzab

The unit vector n̂ is perpendicular to both a = (5, 2, −3) and b = (0, −4, −1) and has a positive z-component. Find its y-component n̂y.

a = (5, 2, −3)b = (0, −4, −1)n̂z > 0
n̂y =
6.Hard
xyzab

Find the area A of the parallelogram spanned by a = (1, −1, 2) and b = (3, 1, −1).

a = (1, −1, 2)b = (3, 1, −1)
A =

Answer key

  1. 1. ûz = −0.5494
  2. 2. Fx = 15 N
  3. 3. θ = 99.21 °
  4. 4. θ = 93.18 °
  5. 5. n̂y = −0.2006
  6. 6. A = 8.124

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