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
xy160°205°F1​F2​

Two forces act at a point in the xy-plane: F₁ = 180 N at 160° and F₂ = 60 N at 205°, 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 = 180 N, θ1 = 160°F2 = 60 N, θ2 = 205°
φ = °
2.Easy
xyza

Find the y-component ûy of the unit vector û that points in the direction of a = (4, −2, −7).

a = (4, −2, −7)
ûy =
3.Medium
xyzab

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

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

Find the angle θ between the vectors a = (4, 4, −1) and b = (−2, 6, −6), in degrees.

a = (4, 4, −1)b = (−2, 6, −6)
θ = °
5.Hard
xyzabc

Find the volume V of the parallelepiped whose edges from one corner are a = (5, −4, 1), b = (1, −4, 0) and c = (3, 2, −3).

a = (5, −4, 1)b = (1, −4, 0)c = (3, 2, −3)
V =
6.Hard
xyzab

For a = (−4, −1, 1) and b = (−3, 2, 5), find the z-component (a × b)z of the cross product a × b.

a = (−4, −1, 1)b = (−3, 2, 5)
(a × b)z =

Answer key

  1. 1. φ = 170.8 °
  2. 2. ûy = −0.2408
  3. 3. a·b = −42
  4. 4. θ = 63.94 °
  5. 5. V = 62
  6. 6. (a × b)z = −11

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