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
xy95°5°F1​F2​

Two forces act at a point in the xy-plane: F₁ = 60 N at 95° and F₂ = 50 N at 5°, both measured anticlockwise from the positive x-axis. Find the magnitude R of their resultant.

F1 = 60 N, θ1 = 95°F2 = 50 N, θ2 = 5°
R = N
2.Easy
xy80°F = 430 N

A force of 430 N acts in the xy-plane at 80° measured anticlockwise from the positive x-axis. Find its x-component Fx.

F = 430 Nθ = 80°
Fx = N
3.Medium
xyzab

Find the dot product a·b of a = (4, −2, −2) and b = (0, −5, 1).

a = (4, −2, −2)b = (0, −5, 1)
a·b =
4.Medium
xyzdFAB

A constant force F = (10, 10, −20) N acts on an object while it moves in a straight line from point A(−3, 4, −5) to point B(1, −4, 0). Coordinates are in metres. Find the work W done by the force.

F = (10, 10, −20) NA(−3, 4, −5) mB(1, −4, 0) m
W = J
5.Hard
xyzab

The unit vector n̂ is perpendicular to both a = (2, 2, 5) and b = (0, −2, −1) and has a positive z-component. Find its y-component n̂y.

a = (2, 2, 5)b = (0, −2, −1)n̂z > 0
n̂y =
6.Hard
xyzab

The unit vector n̂ is perpendicular to both a = (5, 0, −3) and b = (−4, 2, 1) and has a positive z-component. Find its x-component n̂x.

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

Answer key

  1. 1. R = 78.1 N
  2. 2. Fx = 74.67 N
  3. 3. a·b = 8
  4. 4. W = −140 J
  5. 5. n̂y = −0.2182
  6. 6. n̂x = 0.4411

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