Lesson Worksheet: Cross Product in 3D Mathematics

In this worksheet, we will practice finding the cross product of two vectors in space and how to use it to find the area of geometric shapes.


Find |×|ABAB.

  • Atan𝜃
  • Bsin𝜃
  • C0
  • D1
  • Ecos𝜃


Given that A=5,9,1 and B=2,1,7, find AB×.

  • A62+37+23ijk
  • B64+3313ijk
  • C6237+23ijk
  • D23+6237ijk


Let Vi= and Wijk=3+2+4. Calculate VW×.

  • A0,4,2
  • B2,0,0
  • C3,0,0
  • D2,3,0
  • E4,0,3


Find the unit vectors that are perpendicular to both A=4,2,0 and B=4,6,4.

  • A1,2,2 or 1,2,2
  • B23,23,13 or 23,23,13
  • C13,23,23 or 13,23,23
  • D8,16,16 or 8,16,16
  • E0,22,22 or 0,22,22


If A=3,4,4, B=2,5,4, and C=4,4,2, find ()×()ABCA.

  • A6+6+15ijk
  • B228ijk
  • C2+16+19ijk
  • D6615ijk


If A=2,2,1, B=4,4,5, and C=4,2,4, find (×)(×)ABAC.


If A and B are unit vectors and 𝜃 the measure of the angle between them, find |()×(+)|ABAB.

  • Asin𝜃
  • B2𝜃sin
  • C𝐴𝐵𝜃sin
  • D2𝐴𝐵𝜃sin
  • E𝐴𝐵𝜃sin


Find the value of |×|+||2||||ABABAB.

  • A14
  • B2
  • C1
  • D12
  • E0


Given that 𝐷=0,2,8, 𝐸=6,4,6, and 𝐹=4,9,2, determine the area of the triangle 𝐷𝐸𝐹 approximated to the nearest hundredth.


𝐴𝐵𝐶𝐷 is a parallelogram with 𝐴𝐵=1,1,3 and 𝐴𝐷=3,4,1. Find the area of 𝐴𝐵𝐶𝐷. Give your answer to one decimal place.

This lesson includes 24 additional questions and 153 additional question variations for subscribers.

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