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 z-component ûz of the unit vector û that points in the direction of a = (−8, −4, −1).

a = (−8, −4, −1)
ûz =
2.Easy
xya

Find the magnitude |a| of the vector a = (7, −4).

a = (7, −4)
|a| =
3.Medium
xyzab

Find the dot product a·b of a = (6, −2, −5) and b = (−7, −4, −3).

a = (6, −2, −5)b = (−7, −4, −3)
a·b =
4.Medium
xyzab

Find the value of k for which the vectors a = (−5, −5, 2) and b = (3, 1, k) are perpendicular.

a = (−5, −5, 2)b = (3, 1, k)
k =
5.Hard
xyzrFOA

A force F = (−20, −20, −60) N acts at point A(−4, 3, −3). Coordinates are in metres. Find the x-component Mx of the moment of the force about the origin O(0, 0, 0).

F = (−20, −20, −60) NA(−4, 3, −3) mO(0, 0, 0) m
Mx = N·m
6.Hard
xyzabc

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

a = (−1, −1, −1)b = (−3, −5, 0)c = (−1, −4, −5)
V =

Answer key

  1. 1. ûz = −0.1111
  2. 2. |a| = 8.062
  3. 3. a·b = −19
  4. 4. k = 10
  5. 5. Mx = −240 N·m
  6. 6. V = 17

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