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−9−5−7ijk
  • B−11−4−20ijk
  • C−−11+19ijk
  • D−9−11−20ijk

Q2:

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

  • A−8,25,−185
  • B8,25,−185
  • C85,25,−185

Q3:

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

  • A0,5,−52
  • B3,2,−112
  • C⟨10,0,−15⟩
  • D5,0,−152

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||−||VW
  • C|−|VW

Q5:

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

  • A−73+133ij
  • B−73−133ij
  • C253−133+163ijk
  • D−73−13ij

Q6:

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

  • A3+3ik
  • Bijk−2+3
  • Cij−2
  • Dijk−2+5
  • E3+3ij

Q7:

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

  • A1
  • B2
  • C2√3
  • D3√2
  • E2√2

Q8:

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

  • Ai
  • B8−9jk
  • Cijk−8+9
  • D2−8+9ijk

Q9:

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

  • A35,−35
  • B√155,−√155
  • C√15,−√15
  • D15,−15

Q10:

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

  • A116,−116
  • B116121,−116121
  • C2√2911,−2√2911
  • D2√29,−2√29

Q11:

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

  • A√4141(−2+6−)ijk
  • B√8585(−2+9)ik
  • C√721(−2+6−)ijk
  • D√4141(2−6+)ijk

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,−75√3,15√3
  • D1√3,75√3,−15√3

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