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
xy110°315°F1​F2​

Two forces act at a point in the xy-plane: F₁ = 80 N at 110° and F₂ = 120 N at 315°, both measured anticlockwise from the positive x-axis. Find the magnitude R of their resultant.

F1 = 80 N, θ1 = 110°F2 = 120 N, θ2 = 315°
R = N
2.Easy
xyza

Find the y-component ûy of the unit vector û that points in the direction of a = (−8, −2, 5).

a = (−8, −2, 5)
ûy =
3.Medium
xyzab

Find the dot product a·b of a = (6, 0, 4) and b = (4, 6, 0).

a = (6, 0, 4)b = (4, 6, 0)
a·b =
4.Medium
xyzab

Find the dot product a·b of a = (4, 5, −4) and b = (−6, 4, −6).

a = (4, 5, −4)b = (−6, 4, −6)
a·b =
5.Hard
xyzabc

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

a = (1, 3, −3)b = (−2, −2, 4)c = (1, −5, 5)
V =
6.Hard
xyzabc

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

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

Answer key

  1. 1. R = 58.3 N
  2. 2. ûy = −0.2074
  3. 3. a·b = 24
  4. 4. a·b = 20
  5. 5. V = 16
  6. 6. V = 12

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