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 magnitude |a| of the vector a = (2, 9, −3).

a = (2, 9, −3)
|a| =
2.Easy
xyza

Find the z-component ûz of the unit vector û that points in the direction of a = (−6, 8, 6).

a = (−6, 8, 6)
ûz =
3.Medium
xyzab

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

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

Find the scalar projection compb a of a = (−3, −7, −5) onto b = (6, 1, 2), that is, the component of a in the direction of b.

a = (−3, −7, −5)b = (6, 1, 2)
compb a =
5.Hard
xyzab

For a = (3, −6, 2) and b = (3, 6, 6), find the x-component (a × b)x of the cross product a × b.

a = (3, −6, 2)b = (3, 6, 6)
(a × b)x =
6.Hard
xyzabc

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

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

Answer key

  1. 1. |a| = 9.695
  2. 2. ûz = 0.5145
  3. 3. a·b = 30
  4. 4. compb a = −5.466
  5. 5. (a × b)x = −48
  6. 6. V = 48

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