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°315°F1​F2​

Two forces act at a point in the xy-plane: F₁ = 190 N at 80° and F₂ = 110 N at 315°, 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 = 190 N, θ1 = 80°F2 = 110 N, θ2 = 315°
φ = °
2.Easy
xy265°F = 100 N

A force of 100 N acts in the xy-plane at 265° measured anticlockwise from the positive x-axis. Find its x-component Fx.

F = 100 Nθ = 265°
Fx = N
3.Medium
xyzab

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

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

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

a = (−3, 5, −3)b = (k, −3, 6)
k =
5.Hard
xyzABC

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

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

A force F = (−50, −10, 40) N acts at point A(−2, 2, −2). Coordinates are in metres. Find the y-component My of the moment of the force about the origin O(0, 0, 0).

F = (−50, −10, 40) NA(−2, 2, −2) mO(0, 0, 0) m
My = N·m

Answer key

  1. 1. φ = 44.62 °
  2. 2. Fx = −8.716 N
  3. 3. θ = 94.8 °
  4. 4. k = −11
  5. 5. A = 12.21 m²
  6. 6. My = 180 N·m

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