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
xy5°225°F1​F2​

Two forces act at a point in the xy-plane: F₁ = 190 N at 5° and F₂ = 90 N at 225°, 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 = 5°F2 = 90 N, θ2 = 225°
φ = °
2.Easy
xyza

Find the y-component ûy of the unit vector û that points in the direction of a = (−7, 7, 8).

a = (−7, 7, 8)
ûy =
3.Medium
xyzab

Find the dot product a·b of a = (−4, −7, 3) and b = (−6, −2, 3).

a = (−4, −7, 3)b = (−6, −2, 3)
a·b =
4.Medium
xyzab

Find the dot product a·b of a = (3, −4, 6) and b = (−2, −3, −3).

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

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

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

For a = (3, 3, −3) and b = (3, 3, 6), find the x-component (a × b)x of the cross product a × b.

a = (3, 3, −3)b = (3, 3, 6)
(a × b)x =

Answer key

  1. 1. φ = 339.5 °
  2. 2. ûy = 0.55
  3. 3. a·b = 47
  4. 4. a·b = −12
  5. 5. A = 15.23 m²
  6. 6. (a × b)x = 27

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