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
xy105°F = 30 N

A force of 30 N acts in the xy-plane at 105° measured anticlockwise from the positive x-axis. Find its x-component Fx.

F = 30 Nθ = 105°
Fx = N
2.Easy
xyza

Find the magnitude |a| of the vector a = (5, −4, −3).

a = (5, −4, −3)
|a| =
3.Medium
xyzab

Find the angle θ between the vectors a = (−1, −1, −1) and b = (0, 1, −4), in degrees.

a = (−1, −1, −1)b = (0, 1, −4)
θ = °
4.Medium
xyzab

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

a = (−7, −3, −6)b = (−6, −5, −3)
a·b =
5.Hard
xyzab

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

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

For a = (−3, 4, 2) and b = (4, 2, 0), find the x-component (a × b)x of the cross product a × b.

a = (−3, 4, 2)b = (4, 2, 0)
(a × b)x =

Answer key

  1. 1. Fx = −7.765 N
  2. 2. |a| = 7.071
  3. 3. θ = 65.16 °
  4. 4. a·b = 75
  5. 5. n̂x = −0.1456
  6. 6. (a × b)x = −4

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