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

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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.

Q1:

Find |×|⋅ABAB.

  • Atan𝜃
  • Bsin𝜃
  • C0
  • D1
  • Ecos𝜃

Q2:

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

  • A62+37+23ijk
  • B64+33−13ijk
  • C62−37+23ijk
  • D23+62−37ijk

Q3:

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

  • A⟨0,−4,2⟩
  • B⟨2,0,0⟩
  • C⟨3,0,0⟩
  • D⟨−2,3,0⟩
  • E⟨4,0,−3⟩

Q4:

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

  • A⟨−1,2,2⟩ or ⟨1,−2,−2⟩
  • B23,23,−13 or −23,−23,13
  • C−13,23,23 or 13,−23,−23
  • D⟨−8,16,16⟩ or ⟨8,−16,−16⟩
  • E0,−√22,−√22 or 0,√22,√22

Q5:

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

  • A6+6+15ijk
  • B−2−2−8ijk
  • C−2+16+19ijk
  • D−6−6−15ijk

Q6:

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

Q7:

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

  • Asin𝜃
  • B2𝜃sin
  • C𝐴𝐵𝜃sin
  • D2𝐴𝐵𝜃sin
  • E𝐴𝐵𝜃sin

Q8:

Find the value of |×|+|⋅|2||||ABABAB.

  • A14
  • B2
  • C1
  • D12
  • E0

Q9:

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.

Q10:

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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