Worksheet: Vector Operations in 3D

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In this worksheet, we will practice doing operations on vectors in 3D, such as addition, subtraction, and scalar multiplication.

Q1:

If Aijk=−5−8+6 and Bijk=4−3+13, find AB−.

  • A − 9 − 5 − 7 i j k
  • B − 1 1 − 4 − 2 0 i j k
  • C − − 1 1 + 1 9 i j k
  • D − 9 − 1 1 − 2 0 i j k

Q2:

If A=⟨−8,9,9⟩ and B=⟨−6,4,9⟩, find 25−45AB.

  • A − 8 , 2 5 , − 1 8 5
  • B 8 , 2 5 , − 1 8 5
  • C 8 5 , 2 5 , − 1 8 5

Q3:

If A=⟨−1,1,1⟩, and B=⟨1,1,−2⟩, determine the vector C for which 2+5=5CAB.

  • A 0 , 5 , − 5 2
  • B 3 , 2 , − 1 1 2
  • C ⟨ 1 0 , 0 , − 1 5 ⟩
  • D 5 , 0 , − 1 5 2

Q4:

V and W are two vectors, where V=⟨−1,5,−2⟩ and W=⟨3,1,1⟩. Comparing |−|VW and ||−||VW, which quantity is larger?

  • AThey are both the same.
  • B | | − | | V W
  • C | − | V W

Q5:

If Aijk=9−7+8 and Bijk=8−3+4, find AB−23.

  • A − 7 3 + 1 3 3 i j
  • B − 7 3 − 1 3 3 i j
  • C 2 5 3 − 1 3 3 + 1 6 3 i j k
  • D − 7 3 − 1 3 i j

Q6:

If Aijk=++, Bijk=−2+3, and Cijk=−−+, find ABC+–.

  • A 3 + 3 i k
  • B i j k − 2 + 3
  • C i j − 2
  • D i j k − 2 + 5
  • E 3 + 3 i j

Q7:

If Aijk=−−+2, Bjk=+, and Cik=−−, find |––|ABC.

  • A1
  • B2
  • C 2 √ 3
  • D 3 √ 2
  • E 2 √ 2

Q8:

Suppose that A=⟨4,7,−7⟩, B=⟨−5,1,−2⟩, and ABCi++=. What is C?

  • A i
  • B 8 − 9 j k
  • C i j k − 8 + 9
  • D 2 − 8 + 9 i j k

Q9:

Given that Aijk=3++𝑚 and that B is a unit vector equal to 15A, determine the possible values of 𝑚.

  • A 3 5 , − 3 5
  • B √ 1 5 5 , − √ 1 5 5
  • C √ 1 5 , − √ 1 5
  • D 1 5 , − 1 5

Q10:

Suppose a unit vector A is such that 11=⟨−1,−2,𝑘⟩A. Determine the possible values of 𝑘.

  • A 1 1 6 , − 1 1 6
  • B 1 1 6 1 2 1 , − 1 1 6 1 2 1
  • C 2 √ 2 9 1 1 , − 2 √ 2 9 1 1
  • D 2 √ 2 9 , − 2 √ 2 9

Q11:

Given Ajk=−3+5 and Bijk=−2+3+4, determine the unit vector in the direction B–A.

  • A √ 4 1 4 1 ( − 2 + 6 − ) i j k
  • B √ 8 5 8 5 ( − 2 + 9 ) i k
  • C √ 7 2 1 ( − 2 + 6 − ) i j k
  • D √ 4 1 4 1 ( 2 − 6 + ) i j k

Q12:

Find the unit vector in the direction of 𝐴𝐵, where 𝐴⟨−4,5,2⟩ and 𝐵⟨−9,−2,3⟩.

  • A ⟨ − 5 , − 7 , 1 ⟩
  • B ⟨ 1 , 1 , 1 ⟩
  • C − 1 √ 3 , − 7 5 √ 3 , 1 5 √ 3
  • D 1 √ 3 , 7 5 √ 3 , − 1 5 √ 3

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