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
xy80°15°F1​F2​

Two forces act at a point in the xy-plane: F₁ = 290 N at 80° and F₂ = 150 N at 15°, both measured anticlockwise from the positive x-axis. Find the direction φ of their resultant, measured anticlockwise from the positive x-axis (between 0° and 360°).

F1 = 290 N, θ1 = 80°F2 = 150 N, θ2 = 15°
φ = °
2.Easy
xy155°260°F1​F2​

Two forces act at a point in the xy-plane: F₁ = 140 N at 155° and F₂ = 210 N at 260°, both measured anticlockwise from the positive x-axis. Find the magnitude R of their resultant.

F1 = 140 N, θ1 = 155°F2 = 210 N, θ2 = 260°
R = N
3.Medium
xyzab

Find the value of k for which the vectors a = (−1, 5, 4) and b = (0, 2, k) are perpendicular.

a = (−1, 5, 4)b = (0, 2, k)
k =
4.Medium
xyzdFAB

A constant force F = (−20, −5, 20) N acts on an object while it moves in a straight line from point A(−4, −3, −2) to point B(−1, −5, −2). Coordinates are in metres. Find the work W done by the force.

F = (−20, −5, 20) NA(−4, −3, −2) mB(−1, −5, −2) m
W = J
5.Hard
xyzab

Find the area A of the parallelogram spanned by a = (1, 3, −5) and b = (−4, 0, 1).

a = (1, 3, −5)b = (−4, 0, 1)
A =
6.Hard
xyzab

The unit vector n̂ is perpendicular to both a = (0, −4, −2) and b = (−5, 3, −2) and has a positive z-component. Find its x-component n̂x.

a = (0, −4, −2)b = (−5, 3, −2)n̂z > 0
n̂x =

Answer key

  1. 1. φ = 58.96 °
  2. 2. R = 220.2 N
  3. 3. k = −2.5
  4. 4. W = −50 J
  5. 5. A = 22.67
  6. 6. n̂x = −0.5307

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