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×.

  • A i j k − 7 2 + 1 5
  • B 1 3 − 5 4 + 2 1 i j k
  • C − 2 7 − 2 − 2 1 i j k
  • D 2 1 − 5 4 + 1 3 i j k

Q3:

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

  • A 6 2 + 3 7 + 2 3 i j k
  • B 6 4 + 3 3 − 1 3 i j k
  • C 6 2 − 3 7 + 2 3 i j k
  • D 2 3 + 6 2 − 3 7 i j k

Q4:

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

  • A 8 0 + 4 8 j k
  • B 1 6 0 + 9 6 j k
  • C 2 4 − 7 2 − 2 0 0 i j k
  • D − 1 6 0 − 9 6 j k

Q5:

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

  • A 1 6 + 6 − 3 6 i j k
  • B − 3 5 + 2 0 + 4 i j k
  • C 1 9 − 5 2 + 2 0 i j k
  • D 2 0 − 5 2 + 1 9 i j k

Q6:

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

  • A ⟨ 4 , 2 , − 1 2 ⟩
  • 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 , 1 2 , 3 ⟩
  • B ⟨ − 3 , 6 , 9 ⟩
  • C ⟨ 1 2 , 0 , 1 2 ⟩
  • D ⟨ 0 , 6 , − 1 2 ⟩
  • E ⟨ 0 , 0 , 0 ⟩

Q8:

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

  • A ⟨ 1 4 , 1 2 , − 4 0 ⟩
  • B ⟨ − 3 4 , 5 4 , − 6 4 ⟩
  • C ⟨ − 5 2 , − 8 , 3 8 ⟩
  • D ⟨ − 2 8 , 4 , − 4 0 ⟩
  • E ⟨ 6 8 , − 4 8 , 3 8 ⟩

Q9:

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

  • A 6 + 6 + 1 5 i j k
  • B − 2 − 2 − 8 i j k
  • C − 2 + 1 6 + 1 9 i j k
  • D − 6 − 6 − 1 5 i j k

Q10:

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

  • A − 1 3 , 2 3 , 2 3 or 13,−23,−23
  • B ⟨ − 2 4 , 4 8 , 4 8 ⟩ or ⟨24,−48,−48⟩
  • C ⟨ − 1 , 2 , 2 ⟩ or ⟨1,−2,−2⟩
  • D ⟨ − 8 , 1 6 , 1 6 ⟩ 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 , − 2 3 , − 2 4 ⟩
  • B ⟨ 1 1 , − 2 3 , 1 6 ⟩
  • C ⟨ 2 0 , − 4 , − 6 ⟩
  • D ⟨ 2 3 , 2 6 , 4 ⟩
  • E ⟨ − 3 5 , 1 , 1 6 ⟩

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.

  • A 4 √ 1 4 6 − 3 √ 1 4 6 + 1 1 √ 1 4 6 i j k
  • B 4 √ 3 8 6 − 3 √ 3 8 6 + 1 9 √ 3 8 6 i j k
  • C 4 √ 1 4 6 + 3 √ 1 4 6 + 1 1 √ 1 4 6 i j k
  • D 1 1 √ 3 8 6 + 4 √ 3 8 6 + 3 √ 3 8 6 i j k

Q13:

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

  • A ⟨ 7 , 6 , − 2 0 ⟩
  • B ⟨ − 3 4 , 2 4 , − 1 9 ⟩
  • C ⟨ − 1 2 , − 3 6 , − 1 0 ⟩
  • D ⟨ 2 6 , 4 , − 1 9 ⟩
  • E ⟨ − 3 2 , 2 7 , − 1 7 ⟩

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 into B and the area of the triangle drawn.

  • A A B k × = 2 0 4 , area =102 square units
  • B A B k × = 5 2 , area =26 square units
  • C A B k × = 3 3 6 , area =168 square units
  • D A B k × = − 8 8 , area =44 square units
  • E A B k × = 2 7 2 , 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 − 1 1 + 2 + 7 i j k
  • B − − 1 5 − 5 i j k
  • C 2 + 2 4 + 1 0 i j k
  • D i j k + 1 2 + 5

Q21:

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

  • A 2 5 √ 2 0 1
  • B 2 5 √ 1 2 9
  • C 1 0 √ 1 2 9
  • D 1 0 √ 2 0 1

Q22:

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

  • A 1 6 − 1 4 + 2 4 i j k
  • B − 1 6 + 1 4 − 2 4 i j k
  • C 8 − 1 6 − 5 i j k
  • D 8 + 1 6 − 5 i j k

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.

  • A s i n 𝜃
  • B 2 𝜃 s i n
  • C 𝐴 𝐵 𝜃 s i n
  • D 2 𝐴 𝐵 𝜃 s i n
  • E 𝐴 𝐵 𝜃 s i n

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