Worksheet: Vector Triple Product

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In this worksheet, we will practice calculating the triple vector product of three vectors.

Q1:

U , V , and W are three vectors, where U=⟨1,0,2⟩, V=⟨−1,0,3⟩, and W=⟨2,0,−2⟩. Calculate UVW⋅(×) and UVW×(×).

  • A U V W ⋅ ( × ) = 0 , U V W × ( × ) = ⟨ − 8 , 0 , 4 ⟩
  • B U V W ⋅ ( × ) = − 1 4 , U V W × ( × ) = ⟨ − 7 , 0 , 1 ⟩
  • C U V W ⋅ ( × ) = 0 , U V W × ( × ) = ⟨ 1 3 , 0 , − 1 9 ⟩
  • D U V W ⋅ ( × ) = − 5 , U V W × ( × ) = ⟨ − 2 , 0 , − 1 2 ⟩
  • E U V W ⋅ ( × ) = − 1 4 , U V W × ( × ) = ⟨ 1 3 , 0 , − 1 9 ⟩

Q2:

Find ⟨−4,−5,1⟩×⟨0,−5,−5⟩×⟨−1,5,2⟩.

  • A − 6 0 − 8 0 + 1 3 0 i j k
  • B 1 2 5 − 1 0 j k
  • C 3 0 − 2 0 + 2 0 i j k
  • D − 1 4 0 − 8 0 + 1 3 0 i j k

Q3:

If Aij=−−5, Bij=−6−3, and Cij=3+3, determine (×)×BCA.

  • A − 4 5 + 9 i j
  • B − 3 6 + 7 2 i j
  • C − 8 1 + 8 1 i j
  • D 3 6 − 7 2 i j

Q4:

Determine ACB×(×) if Aij=−5+5, Bij=−4+3, and Cij=2+2.

  • A − 7 0 − 7 0 i j
  • B − 1 0 + 1 0 i j
  • C 6 0 + 8 0 i j
  • D 7 0 + 7 0 i j

Q5:

U , V , and W are three vectors, where U=⟨1,1,1⟩, V=⟨3,0,2⟩, and W=⟨2,2,2⟩. Calculate UVW⋅(×) and UVW×(×).

  • A U V W ⋅ ( × ) = 1 0 , U V W × ( × ) = ⟨ 8 , − 1 0 , 2 ⟩
  • B U V W ⋅ ( × ) = 1 0 , U V W × ( × ) = ⟨ 2 8 , 1 0 , 2 2 ⟩
  • C U V W ⋅ ( × ) = 0 , U V W × ( × ) = ⟨ 2 8 , 1 0 , 2 2 ⟩
  • D U V W ⋅ ( × ) = 8 , U V W × ( × ) = ⟨ 6 , 0 , 4 ⟩
  • E U V W ⋅ ( × ) = 0 , U V W × ( × ) = ⟨ 8 , − 1 0 , 2 ⟩

Q6:

Determine the volume of a parallelepiped on the vectors 𝑂𝐴, 𝑂𝐵, and 𝑂𝐶, given that the coordinates of points 𝑂, 𝐴, 𝐵, and 𝐶 are (3,5,3), (−4,0,−2), (−2,−4,0), and (0,5,3), respectively.

Q7:

If Aij=8−9, Bij=−2+2, and Cij=−6+5, determine (×)×ACB.

  • A 2 8 + 2 8 i j
  • B − 1 8 − 1 6 i j
  • C 1 0 + 1 2 i j
  • D 1 8 + 1 6 i j

Q8:

If Aij=−2, Bij=4−6, and Cij=4−5, determine (×)×BAC.

  • A − 1 0 − 8 i j
  • B 1 8 + 1 2 i j
  • C 8 + 4 i j
  • D − 1 8 − 1 2 i j

Q9:

Determine ABC×(×) if Aij=2+6, Bij=−2−4, and Cij=3+8.

  • A 2 4 − 8 i j
  • B − 8 + 4 i j
  • C − 3 2 + 1 2 i j
  • D − 2 4 + 8 i j

Q10:

Find ⟨3,3,4⟩×⟨4,4,1⟩×⟨0,−5,4⟩.

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

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