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Worksheet: Dot Product in 3D

Q1:

Given that and , determine .

Q2:

Given that and , determine .

Q3:

Given that and , determine .

Q4:

Given that the coordinates of 𝐴 , 𝐡 , and 𝐢 are ( 2 , βˆ’ 4 , βˆ’ 2 ) , ( βˆ’ 2 , 3 , 3 ) , and ( 4 , 2 , 5 ) , respectively, determine οƒ  𝐴 𝐡 β‹… οƒ  𝐴 𝐢 .

Q5:

Given that the coordinates of 𝐴 , 𝐡 , and 𝐢 are ( βˆ’ 2 , βˆ’ 4 , 2 ) , ( 5 , βˆ’ 5 , 3 ) , and ( βˆ’ 3 , βˆ’ 2 , 5 ) , respectively, determine οƒ  𝐴 𝐡 β‹… οƒ  𝐴 𝐢 .

Q6:

If and are two perpendicular vectors, then .

Q7:

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.

Q8:

If V i j k = βˆ’ 3 βˆ’ 2 βˆ’ and W i j k = 6 + 4 + 2 , calculate V W β‹… .

Q9:

Find ⟨ 1 , 2 , 3 , 4 ⟩ β‹… ⟨ 2 , 0 , 1 , 3 ⟩ .

Q10:

and Find A B β‹… .

Q11:

If and , find .

Q12:

Given that , , , and is perpendicular to the vector , determine .

  • A7
  • B
  • C15
  • D

Q13:

If , , and , find the value of .

  • A
  • B
  • C
  • D

Q14:

Determine , given that the scalar product of the two vectors and is 27.

Q15:

For the unit vectors i , j , k , what is k k β‹… ?

Q16:

For the unit vectors , , , what is ?

Q17:

V and W are two vectors, where V = ⟨ 6 , 0 , 4 ⟩ and W = ⟨ 0 , 2 , βˆ’ 1 ⟩ . Is V W βŸ‚ ? Justify your answer.

  • AYes because V W β‹… β‰  0 .
  • BYes because V W β‹… = 0 .
  • CNo because V W β‹… = 0 .
  • DNo because V W β‹… β‰  0 .

Q18:

If and , find .

Q19:

Decide which relation holds between the vectors A = ⟨ βˆ’ 3 , 7 , βˆ’ 8 ⟩ and B = ⟨ βˆ’ 6 , βˆ’ 1 , βˆ’ 1 ⟩ .

  • Aperpendicular
  • Bparallel
  • Cotherwise

Q20:

Decide which relation holds between the vectors A = ⟨ 2 , βˆ’ 5 , βˆ’ 1 ⟩ and B = ⟨ 4 , βˆ’ 1 0 , βˆ’ 2 ⟩ .

  • Aotherwise
  • Bperpendicular
  • Cparallel

Q21:

Decide which relation holds between the vectors A = ⟨ 6 , βˆ’ 1 , βˆ’ 7 ⟩ and B = ⟨ 1 8 , βˆ’ 3 , βˆ’ 2 1 ⟩ .

  • Aotherwise
  • Bperpendicular
  • Cparallel

Q22:

Decide which relation holds between the vectors A = ⟨ βˆ’ 9 , βˆ’ 9 , βˆ’ 2 ⟩ and B = ⟨ βˆ’ 4 5 , βˆ’ 4 5 , βˆ’ 1 0 ⟩ .

  • Aotherwise
  • Bperpendicular
  • Cparallel

Q23:

Given that and , determine .

Q24:

For the unit vectors , , , what is ?

Q25:

Let V = ⟨ 5 , 1 , βˆ’ 2 ⟩ and W = ⟨ 4 , βˆ’ 4 , 3 ⟩ . Calculate V W β‹… .