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
xya

Find the magnitude |a| of the vector a = (−1, −6).

a = (−1, −6)
|a| =
2.Easy
xy80°F = 120 N

A force of 120 N acts in the xy-plane at 80° measured anticlockwise from the positive x-axis. Find its x-component Fx.

F = 120 Nθ = 80°
Fx = N
3.Medium
xyzab

Find the angle θ between the vectors a = (1, 6, 1) and b = (6, −3, −2), in degrees.

a = (1, 6, 1)b = (6, −3, −2)
θ = °
4.Medium
xyzab

Find the value of k for which the vectors a = (1, −1, −1) and b = (−4, 5, k) are perpendicular.

a = (1, −1, −1)b = (−4, 5, k)
k =
5.Hard
xyzab

For a = (−1, 2, −5) and b = (−2, −1, 0), find the x-component (a × b)x of the cross product a × b.

a = (−1, 2, −5)b = (−2, −1, 0)
(a × b)x =
6.Hard
xyzab

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

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

Answer key

  1. 1. |a| = 6.083
  2. 2. Fx = 20.84 N
  3. 3. θ = 108.9 °
  4. 4. k = −9
  5. 5. (a × b)x = −5
  6. 6. n̂x = 0.2592

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