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 = (−5, −4, 5).

a = (−5, −4, 5)
ûz =
2.Easy
xyza

Find the z-component ûz of the unit vector û that points in the direction of a = (7, −4, 2).

a = (7, −4, 2)
ûz =
3.Medium
xyzab

Find the angle θ between the vectors a = (5, 2, −5) and b = (3, 1, 0), in degrees.

a = (5, 2, −5)b = (3, 1, 0)
θ = °
4.Medium
xyzab

Find the dot product a·b of a = (−7, −6, 4) and b = (1, −7, 7).

a = (−7, −6, 4)b = (1, −7, 7)
a·b =
5.Hard
xyzab

For a = (−1, −2, −6) and b = (5, −4, −1), find the z-component (a × b)z of the cross product a × b.

a = (−1, −2, −6)b = (5, −4, −1)
(a × b)z =
6.Hard
xyzabc

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

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

Answer key

  1. 1. ûz = 0.6155
  2. 2. ûz = 0.2408
  3. 3. θ = 42.98 °
  4. 4. a·b = 63
  5. 5. (a × b)z = 14
  6. 6. V = 8

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