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 = (7, −8, 3).

a = (7, −8, 3)
ûy =
2.Easy
xyza

Find the magnitude |a| of the vector a = (4, 7, 5).

a = (4, 7, 5)
|a| =
3.Medium
xyzab

Find the dot product a·b of a = (3, 6, −6) and b = (−1, −7, −7).

a = (3, 6, −6)b = (−1, −7, −7)
a·b =
4.Medium
xyzab

Find the angle θ between the vectors a = (−5, 5, −5) and b = (−3, 6, 0), in degrees.

a = (−5, 5, −5)b = (−3, 6, 0)
θ = °
5.Hard
xyzab

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

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

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

a = (−2, 1, −4)b = (−5, −5, −3)c = (4, −5, −3)
V =

Answer key

  1. 1. ûy = −0.7243
  2. 2. |a| = 9.487
  3. 3. a·b = −3
  4. 4. θ = 39.23 °
  5. 5. n̂y = −0.3123
  6. 6. V = 207

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