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
xy25°115°F1​F2​

Two forces act at a point in the xy-plane: F₁ = 100 N at 25° and F₂ = 250 N at 115°, 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 = 100 N, θ1 = 25°F2 = 250 N, θ2 = 115°
φ = °
2.Easy
xy145°65°F1​F2​

Two forces act at a point in the xy-plane: F₁ = 270 N at 145° and F₂ = 40 N at 65°, both measured anticlockwise from the positive x-axis. Find the magnitude R of their resultant.

F1 = 270 N, θ1 = 145°F2 = 40 N, θ2 = 65°
R = N
3.Medium
xyzab

Find the angle θ between the vectors a = (−2, 5, 6) and b = (−6, −3, 0), in degrees.

a = (−2, 5, 6)b = (−6, −3, 0)
θ = °
4.Medium
xyzab

Find the value of k for which the vectors a = (−5, −6, 0) and b = (3, k, −4) are perpendicular.

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

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

a = (−4, 3, 2)b = (−3, −2, −2)n̂z > 0
n̂y =
6.Hard
xyzrFOA

A force F = (−10, 10, 50) N acts at point A(−4, 3, −4). Coordinates are in metres. Find the z-component Mz of the moment of the force about point O(1, 2, 0).

F = (−10, 10, 50) NA(−4, 3, −4) mO(1, 2, 0) m
Mz = N·m

Answer key

  1. 1. φ = 93.2 °
  2. 2. R = 279.7 N
  3. 3. θ = 93.18 °
  4. 4. k = −2.5
  5. 5. n̂y = −0.6331
  6. 6. Mz = −40 N·m

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