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 x-component ûx of the unit vector û that points in the direction of a = (−6, 3, 7).

a = (−6, 3, 7)
ûx =
2.Easy
xyza

Find the z-component ûz of the unit vector û that points in the direction of a = (−3, 4, 1).

a = (−3, 4, 1)
ûz =
3.Medium
xyzab

Find the scalar projection compb a of a = (5, 3, 1) onto b = (−4, 1, 6), that is, the component of a in the direction of b.

a = (5, 3, 1)b = (−4, 1, 6)
compb a =
4.Medium
xyzab

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

a = (−4, 2, 2)b = (−2, 3, −3)
a·b =
5.Hard
xyzABC

Find the area A of the triangle with corners A(0, −1, 4), B(−3, 1, −3) and C(3, 0, 1). Coordinates are in metres.

A(0, −1, 4)B(−3, 1, −3)C(3, 0, 1)
A = m²
6.Hard
xyzabc

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

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

Answer key

  1. 1. ûx = −0.6189
  2. 2. ûz = 0.1961
  3. 3. compb a = −1.511
  4. 4. a·b = 8
  5. 5. A = 15.67 m²
  6. 6. V = 174

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