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
xyza

Find the y-component ûy of the unit vector û that points in the direction of a = (3, −8, −2).

a = (3, −8, −2)
ûy =
2.Easy
xy60°30°F1​F2​

Two forces act at a point in the xy-plane: F₁ = 190 N at 60° and F₂ = 200 N at 30°, both measured anticlockwise from the positive x-axis. Find the magnitude R of their resultant.

F1 = 190 N, θ1 = 60°F2 = 200 N, θ2 = 30°
R = N
3.Medium
xyzab

Find the scalar projection compb a of a = (5, −2, −5) onto b = (−2, −5, −1), that is, the component of a in the direction of b.

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

Find the scalar projection compb a of a = (−2, 2, 1) onto b = (−4, −2, −2), that is, the component of a in the direction of b.

a = (−2, 2, 1)b = (−4, −2, −2)
compb a =
5.Hard
xyzab

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

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

For a = (−2, 5, −4) and b = (1, 2, −4), find the y-component (a × b)y of the cross product a × b.

a = (−2, 5, −4)b = (1, 2, −4)
(a × b)y =

Answer key

  1. 1. ûy = −0.9117
  2. 2. R = 376.7 N
  3. 3. compb a = 0.9129
  4. 4. compb a = 0.4082
  5. 5. n̂x = −0.5774
  6. 6. (a × b)y = −12

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