Lesson Worksheet: Cross Product in 3D Mathematics • 12th Grade

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 |Γ—|β‹…ABAB.

  • Atanπœƒ
  • Bsinπœƒ
  • C0
  • D1
  • Ecosπœƒ


Given that A=βŸ¨βˆ’5,βˆ’9,βˆ’1⟩ and B=⟨2,βˆ’1,βˆ’7⟩, find ABΓ—.

  • A62+37+23ijk
  • B64+33βˆ’13ijk
  • C62βˆ’37+23ijk
  • D23+62βˆ’37ijk


Let Vi= and Wijk=3+2+4. Calculate VWΓ—.

  • A⟨0,βˆ’4,2⟩
  • B⟨2,0,0⟩
  • C⟨3,0,0⟩
  • DβŸ¨βˆ’2,3,0⟩
  • E⟨4,0,βˆ’3⟩


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

  • AβŸ¨βˆ’1,2,2⟩ or ⟨1,βˆ’2,βˆ’2⟩
  • B23,23,βˆ’13ο‡· or ο‡³βˆ’23,βˆ’23,13ο‡·
  • Cο‡³βˆ’13,23,23ο‡· or 13,βˆ’23,βˆ’23ο‡·
  • DβŸ¨βˆ’8,16,16⟩ or ⟨8,βˆ’16,βˆ’16⟩
  • Eο“’0,βˆ’βˆš22,βˆ’βˆš22ο““ or ο“’0,√22,√22ο““


If A=⟨3,4,βˆ’4⟩, B=⟨2,5,βˆ’4⟩, and C=βŸ¨βˆ’4,βˆ’4,2⟩, find (βˆ’)Γ—(βˆ’)ABCA.

  • A6+6+15ijk
  • Bβˆ’2βˆ’2βˆ’8ijk
  • Cβˆ’2+16+19ijk
  • Dβˆ’6βˆ’6βˆ’15ijk


If A=βŸ¨βˆ’2,2,βˆ’1⟩, B=βŸ¨βˆ’4,βˆ’4,βˆ’5⟩, and C=βŸ¨βˆ’4,2,4⟩, find (Γ—)β‹…(Γ—)ABAC.


If A and B are unit vectors and πœƒ the measure of the angle between them, find |(βˆ’)Γ—(+)|ABAB.

  • Asinπœƒ
  • B2πœƒsin
  • Cπ΄π΅πœƒsin
  • D2π΄π΅πœƒsin
  • Eπ΄π΅πœƒοŠ¨οŠ¨sin


Find the value of |Γ—|+|β‹…|2||||ABABAB.

  • A14
  • B2
  • C1
  • D12
  • E0


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


𝐴𝐡𝐢𝐷 is a parallelogram with 𝐴𝐡=βŸ¨βˆ’1,1,3⟩ and 𝐴𝐷=⟨3,4,1⟩. Find the area of 𝐴𝐡𝐢𝐷. Give your answer to one decimal place.

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

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