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
xy100°150°F1​F2​

Two forces act at a point in the xy-plane: F₁ = 210 N at 100° and F₂ = 270 N at 150°, 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 = 210 N, θ1 = 100°F2 = 270 N, θ2 = 150°
φ = °
2.Easy
xyza

Find the y-component ûy of the unit vector û that points in the direction of a = (−3, −4, 1).

a = (−3, −4, 1)
ûy =
3.Medium
xyzab

Find the dot product a·b of a = (1, −3, 0) and b = (−1, 1, 1).

a = (1, −3, 0)b = (−1, 1, 1)
a·b =
4.Medium
xyzab

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

a = (−4, −4, −5)b = (−1, −5, k)
k =
5.Hard
xyzab

The unit vector n̂ is perpendicular to both a = (−1, −3, −5) and b = (2, −5, −4) and has a positive z-component. Find its x-component n̂x.

a = (−1, −3, −5)b = (2, −5, −4)n̂z > 0
n̂x =
6.Hard
xyzrFOA

A force F = (60, −20, −20) N acts at point A(−4, 2, −2). Coordinates are in metres. Find the magnitude |M| of the moment of the force about the origin O(0, 0, 0).

F = (60, −20, −20) NA(−4, 2, −2) mO(0, 0, 0) m
|M| = N·m

Answer key

  1. 1. φ = 128.3 °
  2. 2. ûy = −0.7845
  3. 3. a·b = −4
  4. 4. k = 4.8
  5. 5. n̂x = −0.5897
  6. 6. |M| = 219.1 N·m

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