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
xya

Find the magnitude |a| of the vector a = (−12, 5).

a = (−12, 5)
|a| =
2.Easy
xy175°70°F1​F2​

Two forces act at a point in the xy-plane: F₁ = 220 N at 175° and F₂ = 290 N at 70°, 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 = 220 N, θ1 = 175°F2 = 290 N, θ2 = 70°
φ = °
3.Medium
xyzab

Find the scalar projection compb a of a = (−4, 1, 5) onto b = (−4, −1, 0), that is, the component of a in the direction of b.

a = (−4, 1, 5)b = (−4, −1, 0)
compb a =
4.Medium
xyzab

Find the dot product a·b of a = (0, −5, −7) and b = (1, −3, −1).

a = (0, −5, −7)b = (1, −3, −1)
a·b =
5.Hard
xyzABC

Find the area A of the triangle with corners A(1, −4, 3), B(4, 0, −3) and C(−3, 1, −1). Coordinates are in metres.

A(1, −4, 3)B(4, 0, −3)C(−3, 1, −1)
A = m²
6.Hard
xyzrFOA

A force F = (−20, −40, −60) N acts at point A(−2, 0, −2). Coordinates are in metres. Find the y-component My of the moment of the force about point O(0, 1, 2).

F = (−20, −40, −60) NA(−2, 0, −2) mO(0, 1, 2) m
My = N·m

Answer key

  1. 1. |a| = 13
  2. 2. φ = 112.4 °
  3. 3. compb a = 3.638
  4. 4. a·b = 22
  5. 5. A = 24.76 m²
  6. 6. My = −40 N·m

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