Worksheet: Dot Product in 3D

In this worksheet, we will practice finding the dot product of two vectors and deducing whether they are perpendicular, parallel, or intersecting.

Q1:

If A and B are two perpendicular vectors, then A B = .

Q2:

If the dot product of two nonzero vectors is zero, what does this tell us about the two vectors?

  • AThey are parallel in opposite directions.
  • BThey are parallel in the same direction.
  • CIt does not indicate anything about the two vectors.
  • DThey are perpendicular.
  • EThe two vectors are equal in magnitude but opposite in direction.

Q3:

For the unit vectors 𝑖 , 𝑗 , 𝑘 , what is 𝑖 𝑖 ?

Q4:

and Find .

Q5:

Given that A = 6 , 3 , 5 and B = 7 , 4 , 1 , determine A B .

Q6:

Given that the coordinates of 𝐴 , 𝐵 , and 𝐶 are ( 2 , 4 , 2 ) , ( 2 , 3 , 3 ) , and ( 4 , 2 , 5 ) , respectively, determine 𝐴 𝐵 𝐴 𝐶 .

Q7:

For what value of 𝑘 are vectors A = 7 , 7 𝑘 , 6 and B = 7 , 3 , 𝑘 perpendicular?

  • A 7 1 5
  • B 4 9 1 5
  • C 7 3
  • D 4 9 1 5

Q8:

Given that 𝐴 𝐵 𝐶 𝐷 is a square of side 33, determine 𝐴 𝐵 𝐶 𝐴 .

Q9:

Find .

Q10:

Given that A i j k = 5 7 + 7 and B i j k = 7 2 5 , determine A B .

Q11:

If A and B are two perpendicular unit vectors, find ( 3 ) ( 2 + ) A B A B .

Q12:

Which of the following is true of the vectors A = 3 , 7 , 8 and B = 6 , 1 , 1 ?

  • AThey are perpendicular.
  • BThey are parallel.
  • CThey are neither parallel nor perpendicular.

Q13:

If 𝑉 = 3 𝑖 2 𝑗 𝑘 and 𝑊 = 6 𝑖 + 4 𝑗 + 2 𝑘 , calculate 𝑉 𝑊 .

Q14:

If A = 0 , 3 , 1 and B i j k = 2 4 , find A B .

Q15:

Given that A i j k = 3 5 + , B i j k = 5 3 3 , C i j k = 2 + 4 , and ( + 𝑚 ) A B is perpendicular to the vector C , determine 𝑚 .

  • A7
  • B 7
  • C15
  • D 1 5

Q16:

If A = 3 i + 3 j + 4 𝑚 k , B = 4 i 6 j 7 k , and A B , find the value of 𝑚 .

  • A 1 5 1 4
  • B 1 7 4
  • C 3 2
  • D 3 1 4

Q17:

Determine 𝑚 , given that the scalar product of the two vectors A i j k = 𝑚 6 6 and B i j k = 2 + 8 4 is 27.

Q18:

For the unit vectors 𝑖 , 𝑗 , 𝑘 , what is 𝑘 𝑘 ?

Q19:

For the unit vectors 𝑖 , 𝑗 , 𝑘 , what is 𝑗 𝑘 ?

Q20:

and are two vectors, where and . Is ? Justify your answer.

  • AYes because .
  • BYes because .
  • CNo because .
  • DNo because .

Q21:

If A i j k = 2 5 3 and B i j k = 4 2 , find A B .

Q22:

Let 𝑉 = ( 5 , 1 , 2 ) and 𝑊 = ( 4 , 4 , 3 ) . Calculate 𝑉 𝑊 .

Q23:

If A i j k = 3 + 2 3 , B i j k = 2 2 , and C i j k = 2 3 , find 2 [ 2 ( 2 + 4 ) ] C A B C B .

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

Q24:

For the unit vectors 𝑖 , 𝑗 , 𝑘 , what is 𝑘 𝑖 ?

Q25:

Given that 5 A = 1 0 i 1 5 j + 1 0 k and 4 B = 8 i + 1 2 j 4 k , determine 4 A 4 B .

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