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.

Practice Means Progress

Download the Nagwa Practice app to access 34 additional questions for this lesson!

scan me!

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.