Lesson Worksheet: Scalar Triple Product Mathematics

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In this worksheet, we will practice calculating the scalar triple product and apply this in geometrical applications.

Q1:

True or False: The scalar triple product of vectors is equal to the determinant of the matrix formed from these vectors.

  • ATrue
  • BFalse

Q2:

Find ijkjkikij⋅(×)+⋅(×)+⋅(×).

Q3:

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

Q4:

Consider ⃑𝐴=(1,2,3), ⃑𝐵=(−4,−5,3), and ⃑𝐶=(1,2,−2).

Find ⃑𝐴⋅⃑𝐵×⃑𝐶.

Find ⃑𝐵⋅⃑𝐴×⃑𝐶.

Q5:

Given that A=⟨1,2,4⟩, B=⟨2,4,−1⟩, and C=⟨−1,4,2⟩, find ABCBCACAB⋅(×)+⋅(×)+⋅(×).

Q6:

If v, w, and u are three linearly independent vectors in ℝ, then the volume of the parallelepiped determined by v, w, and u is .

  • A|⋅(×)|vwu
  • B||+||+||vwu
  • C||+||+||vwu
  • D||⋅||⋅||vwu

Q7:

True or False: The scalar triple product of ⃑𝑢, ⃑𝑣, and ⃑𝑤 results in a vector whose length is equal to the volume of the parallelepiped determined by ⃑𝑢, ⃑𝑣, and ⃑𝑤.

  • ATrue
  • BFalse

Q8:

Find the volume of the parallelepiped with the adjacent sides U=⟨1,1,3⟩, V=⟨2,1,4⟩, and W=⟨5,1,−2⟩.

Q9:

Find the value of 𝑘 for which the four points (1,7,−2), (3,5,6), (−1,6,−4), and (−4,−3,𝑘) all lie in a single plane.

Q10:

The parallelepiped on vectors ⟨−2,−2,𝑚⟩, ⟨2,0,−2⟩, and ⟨−5,1,0⟩ has volume 48. What can 𝑚 be?

  • A16 or −32
  • B16 or −12
  • C36 or −32
  • D36 or −12

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