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 = (−8, −8, −5).

a = (−8, −8, −5)
ûz =
2.Easy
xyza

Find the x-component ûx of the unit vector û that points in the direction of a = (−1, −2, −8).

a = (−1, −2, −8)
ûx =
3.Medium
xyzab

Find the angle θ between the vectors a = (−2, 2, 5) and b = (1, −6, −5), in degrees.

a = (−2, 2, 5)b = (1, −6, −5)
θ = °
4.Medium
xyzdFAB

A constant force F = (−30, −15, −10) N acts on an object while it moves in a straight line from point A(2, 5, 2) to point B(−5, 3, 0). Coordinates are in metres. Find the work W done by the force.

F = (−30, −15, −10) NA(2, 5, 2) mB(−5, 3, 0) m
W = J
5.Hard
xyzABC

Find the area A of the triangle with corners A(−2, −3, 1), B(−4, −2, 4) and C(0, 0, −3). Coordinates are in metres.

A(−2, −3, 1)B(−4, −2, 4)C(0, 0, −3)
A = m²
6.Hard
xyzabc

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

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

Answer key

  1. 1. ûz = −0.4042
  2. 2. ûx = −0.1204
  3. 3. θ = 149.6 °
  4. 4. W = 260 J
  5. 5. A = 7.697 m²
  6. 6. V = 43

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