Worksheet: Cross Product in 3D

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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:

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⟩

Q2:

Given that Aijk=−9−+3 and Bijk=3−2−7, determine AB×.

  • Aijk−72+15
  • B13−54+21ijk
  • C−27−2−21ijk
  • D21−54+13ijk

Q3:

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

Q4:

Given that Aijk=−3+3−5 and Bijk=−−3+5, determine (4)×(2)AB.

  • A80+48jk
  • B160+96jk
  • C24−72−200ijk
  • D−160−96jk

Q5:

If A=⟨4,−2,−9⟩ and B=⟨4,3,4⟩, determine AB×.

  • A16+6−36ijk
  • B−35+20+4ijk
  • C19−52+20ijk
  • D20−52+19ijk

Q6:

V and W are two vectors, where V=⟨2,1,4⟩ and W=⟨1,−2,0⟩. Calculate VW×.

  • A⟨4,2,−12⟩
  • B⟨4,−2,−9⟩
  • C⟨−8,4,3⟩
  • D⟨2,−2,0⟩
  • E⟨8,4,−5⟩

Q7:

V and W are two vectors, where Vijk=−+2+ and Wijk=−3+6+3. Calculate VW×.

  • A⟨3,12,3⟩
  • B⟨−3,6,9⟩
  • C⟨12,0,12⟩
  • D⟨0,6,−12⟩
  • E⟨0,0,0⟩

Q8:

V and W are two vectors, where V=⟨7,2,−10⟩ and W=⟨2,6,4⟩. Calculate VW×.

  • A⟨14,12,−40⟩
  • B⟨−34,54,−64⟩
  • C⟨−52,−8,38⟩
  • D⟨−28,4,−40⟩
  • E⟨68,−48,38⟩

Q9:

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

Q10:

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

  • A−13,23,23 or 13,−23,−23
  • B⟨−24,48,48⟩ or ⟨24,−48,−48⟩
  • C⟨−1,2,2⟩ or ⟨1,−2,−2⟩
  • D⟨−8,16,16⟩ or ⟨8,−16,−16⟩

Q11:

V and W are two vectors, where V=⟨5,1,−2⟩ and W=⟨4,−4,3⟩. Calculate VW×.

  • A⟨−5,−23,−24⟩
  • B⟨11,−23,16⟩
  • C⟨20,−4,−6⟩
  • D⟨23,26,4⟩
  • E⟨−35,1,16⟩

Q12:

Given that A=⟨−3,4,0⟩, and B=⟨1,−5,1⟩, determine the unit vector perpendicular to the plane of the two vectors A and B.

  • A4√146−3√146+11√146ijk
  • B4√386−3√386+19√386ijk
  • C4√146+3√146+11√146ijk
  • D11√386+4√386+3√386ijk

Q13:

Let V=⟨1,3,2⟩ and W=⟨7,2,−10⟩. Calculate VW×.

  • A⟨7,6,−20⟩
  • B⟨−34,24,−19⟩
  • C⟨−12,−36,−10⟩
  • D⟨26,4,−19⟩
  • E⟨−32,27,−17⟩

Q14:

Assuming that (,,)ijk form a right-hand system, Aij=16+4, Bij=19+8, and A and B form two adjacent sides of a triangle, find the vector product of A and B and the area of the triangle drawn.

  • AABk×=204, area =102 square units
  • BABk×=52, area =26 square units
  • CABk×=336, area =168 square units
  • DABk×=−88, area =44 square units
  • EABk×=272, area =136 square units

Q15:

Given that the area of △𝐴𝐵𝐶 is 340, what is ||𝐵𝐴×𝐵𝐶||?

Q16:

If 𝐴𝐵𝐶 is a triangle of area 248.5 cm2, find the value of ||𝐵𝐴×𝐴𝐶||.

Q17:

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.

Q18:

Triangle 𝐴𝐵𝐶 has vertices 𝐴(5,−4), 𝐵(−1,−5), and 𝐶(−3,2). Use vectors to determine its area.

Q19:

Suppose that A=⟨1,1,3⟩ and B=⟨4,8,−8⟩ fix two sides of a triangle. What is the area of this triangle, to the nearest hundredth?

Q20:

If A=⟨−5,0,1⟩ and B=⟨3,1,−3⟩, find AAB×(−2).

  • A−11+2+7ijk
  • B−−15−5ijk
  • C2+24+10ijk
  • Dijk+12+5

Q21:

If Ajk=−10+5 and Bijk=−4+9+, find |5×|BA.

  • A25√201
  • B25√129
  • C10√129
  • D10√201

Q22:

If A=⟨−2,−4,−1⟩, B=⟨−4,1,5⟩, and C=⟨0,3,0⟩, find ABC×(+).

  • A16−14+24ijk
  • B−16+14−24ijk
  • C8−16−5ijk
  • D8+16−5ijk

Q23:

Given that A=⟨9,−5,−1⟩, B=⟨7,−𝑘,−5⟩, C=⟨10,−55,𝑚−3⟩, and 𝐴𝐵∥C, find 𝑘−𝑚.

Q24:

Given that the vectors (6,,1)k and (12,−6,2) are parallel, determine the value of k.

Q25:

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

  • Asin𝜃
  • B2𝜃sin
  • C𝐴𝐵𝜃sin
  • D2𝐴𝐵𝜃sin
  • E𝐴𝐵𝜃sin

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