Worksheet: Scalar Triple Product

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

Q1:

If A = 3 , 0 , 2 , B = 1 , 3 , 3 , and C = 2 , 2 , 1 , find ( + ) [ ( × ) × ( × ) ] A C A B B C .

Q2:

Given A = 1 , 5 , 5 , B = 2 , 4 , 3 , and C = 0 , 5 , 4 , find A B C ( × ) .

Q3:

Find i j k j k i k i j ( × ) + ( × ) + ( × ) .

Q4:

If A i j = 2 + , B i j = 2 4 , and C i j = 9 + 3 , calculate ( × ) A C B .

  • A 3 0 + 6 0 i j
  • B 1 8 5 4 i j
  • C 3 0 k
  • D 3 0 6 0 i j
  • E0

Q5:

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

Q6:

If A i j = 3 + 7 , B i j = 2 2 , and C i j = 4 , calculate ( ) × B C A .

Q7:

Find the volume of the parallelepiped in which three adjacent sides are represented by the vectors A = 3 , 2 , 5 , B = 1 , 7 , 8 , and C = 7 , 2 , 5 .

Q8:

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

  • A16 or 1 2
  • B36 or 1 2
  • C36 or 3 2
  • D16 or 3 2

Q9:

Find the volume of the parallelepiped with the adjacent sides U = 1 , 3 , 2 , V = 7 , 2 , 1 0 , and W = 1 , 0 , 1 .

Q10:

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

  • A | 𝑣 | | 𝑤 | | 𝑢 |
  • B | 𝑣 | + | 𝑤 | + | 𝑢 |
  • C | 𝑣 | + | 𝑤 | + | 𝑢 |
  • D | 𝑣 ( 𝑤 × 𝑢 ) |

Q11:

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.

Q12:

Suppose that vectors ( 3 , 4 , 5 ) , ( 1 , 2 , 3 ) , and ( 3 , 𝑘 , 3 ) are all in the same plane. What is the value of 𝑘 ?

Q13:

Given A = 4 , 5 , 1 , B = 4 , 1 , 5 , and C = 5 , 1 , 1 , find A B C ( × ) .

Q14:

If A i j = 5 5 , B i j = + , and C i j = 8 5 , calculate ( × ) C B A .

  • A 5 0 8 0 i j
  • B0
  • C 3 0 k
  • D 5 0 + 8 0 i j
  • E 1 5 1 5 i j

Q15:

If A i j = 2 + 5 , B i j = + 2 , and C i j = 4 + 9 , calculate ( × ) B A C .

  • A 3 8 k
  • B 8 5 + 3 4 i j
  • C 7 6 + 3 8 i j
  • D0
  • E 7 6 3 8 i j

Q16:

If A i j = 3 , B i j = 3 + 2 , and C i j = 8 + 8 , calculate ( × ) B C A .

  • A 6 4 9 6 i j
  • B 2 4 2 4 i j
  • C0
  • D 3 2 k
  • E 6 4 + 9 6 i j

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