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 = (−4, −4).

a = (−4, −4)
|a| =
2.Easy
xy100°35°F1​F2​

Two forces act at a point in the xy-plane: F₁ = 230 N at 100° and F₂ = 70 N at 35°, 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 = 230 N, θ1 = 100°F2 = 70 N, θ2 = 35°
φ = °
3.Medium
xyzab

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

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

Find the scalar projection compb a of a = (−2, 2, −3) onto b = (−6, 4, 6), that is, the component of a in the direction of b.

a = (−2, 2, −3)b = (−6, 4, 6)
compb a =
5.Hard
xyzABC

Find the area A of the triangle with corners A(−2, 3, 1), B(4, −4, −2) and C(3, −3, 3). Coordinates are in metres.

A(−2, 3, 1)B(4, −4, −2)C(3, −3, 3)
A = m²
6.Hard
xyzrFOA

A force F = (−60, −30, −20) N acts at point A(−4, 4, −3). Coordinates are in metres. Find the y-component My of the moment of the force about point O(−3, −1, −3).

F = (−60, −30, −20) NA(−4, 4, −3) mO(−3, −1, −3) m
My = N·m

Answer key

  1. 1. |a| = 5.657
  2. 2. φ = 86.27 °
  3. 3. k = −3
  4. 4. compb a = 0.2132
  5. 5. A = 20.94 m²
  6. 6. My = −20 N·m

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