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 magnitude |a| of the vector a = (1, 3, −8).

a = (1, 3, −8)
|a| =
2.Easy
xy345°F = 210 N

A force of 210 N acts in the xy-plane at 345° measured anticlockwise from the positive x-axis. Find its x-component Fx.

F = 210 Nθ = 345°
Fx = N
3.Medium
xyzab

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

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

Find the angle θ between the vectors a = (4, 2, 4) and b = (3, −1, 4), in degrees.

a = (4, 2, 4)b = (3, −1, 4)
θ = °
5.Hard
xyzab

The unit vector n̂ is perpendicular to both a = (5, −5, −4) and b = (−5, 1, −5) and has a positive z-component. Find its x-component n̂x.

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

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

a = (−4, 4, −2)b = (5, 4, 0)n̂z > 0
n̂x =

Answer key

  1. 1. |a| = 8.602
  2. 2. Fx = 202.8 N
  3. 3. θ = 73.97 °
  4. 4. θ = 31.81 °
  5. 5. n̂x = −0.5074
  6. 6. n̂x = −0.2094

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