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
xy330°F = 470 N

A force of 470 N acts in the xy-plane at 330° measured anticlockwise from the positive x-axis. Find its x-component Fx.

F = 470 Nθ = 330°
Fx = N
2.Easy
xyza

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

a = (4, 3, −3)
|a| =
3.Medium
xyzab

Find the value of k for which the vectors a = (0, 5, −2) and b = (−2, −4, k) are perpendicular.

a = (0, 5, −2)b = (−2, −4, k)
k =
4.Medium
xyzab

Find the scalar projection compb a of a = (5, −3, 1) onto b = (−2, −1, 4), that is, the component of a in the direction of b.

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

Find the area A of the parallelogram spanned by a = (−5, 1, −3) and b = (5, 1, 0).

a = (−5, 1, −3)b = (5, 1, 0)
A =
6.Hard
xyzabc

Find the volume V of the parallelepiped whose edges from one corner are a = (−2, 1, 3), b = (0, 1, 4) and c = (5, 4, 0).

a = (−2, 1, 3)b = (0, 1, 4)c = (5, 4, 0)
V =

Answer key

  1. 1. Fx = 407 N
  2. 2. |a| = 5.831
  3. 3. k = −10
  4. 4. compb a = −0.6547
  5. 5. A = 18.28
  6. 6. V = 37

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