Lesson Worksheet: Vector Operations in 3D Mathematics

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:

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

Q5:

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

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

Q6:

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

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

Q7:

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

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

Q8:

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

Q9:

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

Q10:

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

Q11:

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

Q12:

Find 𝑘 so that the points (3,9,−4), (9,−3,−1), (−7,29,𝑘) are collinear.

Q13:

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

Q14:

Given the two vectors A=⟨−2,−3,0⟩ and B=⟨−3,3,−2⟩, find AB+.

  • A⟨−5,0,2⟩
  • B−7
  • C⟨−5,0,−2⟩
  • D⟨1,0,−2⟩
  • E⟨1,−6,2⟩

Q15:

If A=⟨2,3,4⟩ and 𝐶=2, find 𝐶A.

  • A⟨−4,6,−8⟩
  • B⟨4,−6,8⟩
  • C⟨4,6,8⟩
  • D⟨2,−6,4⟩
  • E⟨−4,−6,−8⟩

Q16:

Given that A=⟨1,2,1⟩ and B=⟨2,1,2⟩, find |+|AB.

  • A3√3
  • B3
  • C√13
  • D9
  • E√3

Q17:

If A=⟨2,3,1⟩ and B=⟨1,3,2⟩, find 2+3AB.

  • A⟨7,11,2⟩
  • B⟨2,11,1⟩
  • C⟨7,15,8⟩
  • D⟨1,15,2⟩
  • E⟨6,11,6⟩

Q18:

True or False: If A=⟨1,1,2⟩ and B=⟨2,1,2⟩, then |+|>|−|ABAB.

  • AFalse
  • BTrue

Q19:

True or False: Given that A=⟨1,2,3⟩, B=⟨2,6,2⟩, and C=⟨0,2,0⟩, then |++|>|−−|ABCABC.

  • ATrue
  • BFalse

Q20:

Given that A=⟨1,2,2⟩ and B=⟨4,3,1⟩, find the vector C that satisfies the equation CAB=2+.

  • A⟨6,4,5⟩
  • B⟨4,7,2⟩
  • C⟨6,7,5⟩
  • D⟨4,7,1⟩
  • E⟨4,4,2⟩

Q21:

A force of (−++)ijk newtons is being applied to an object. What other force should be applied to achieve a total force of (2++)ijk newtons?

  • A(−−2−2)ijk newtons
  • B−3i newtons
  • C(+2+2)ijk newtons
  • D(3++)ijk newtons
  • E3i newtons

Q22:

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

  • A|+|VW
  • B||+||VW
  • CThey are both the same.

Q23:

Given that A=⟨𝑢,𝑢,𝑢⟩ and B=⟨𝑣,𝑣,𝑣⟩, find the following.

AB+

  • A⟨𝑢+𝑣,𝑢+𝑣,𝑢+𝑣⟩
  • B⟨𝑢−𝑣,𝑢−𝑣,𝑢−𝑣⟩
  • C⟨𝑢𝑣,𝑢𝑣,𝑢𝑣⟩
  • D⟨𝑢+𝑣,𝑢+𝑣,𝑢+𝑣⟩
  • E⟨𝑢𝑣,𝑢𝑣,𝑢𝑣⟩

AB+𝑘, where 𝑘 is a constant

  • A⟨𝑘𝑢+𝑣,𝑘𝑢+𝑣,𝑘𝑢+𝑣⟩
  • B⟨𝑘𝑢𝑣,𝑘𝑢𝑣,𝑘𝑢𝑣⟩
  • C⟨𝑢+𝑘𝑣,𝑢+𝑘𝑣,𝑢+𝑘𝑣⟩
  • D⟨𝑢+𝑣,𝑢+𝑣,𝑢+𝑣⟩
  • E⟨𝑘𝑢𝑣,𝑘𝑢𝑣,𝑘𝑢𝑣⟩

|+|AB

  • A(𝑢+𝑣)+(𝑢+𝑣)+(𝑢+𝑣)
  • B(𝑢+𝑢)+(𝑣+𝑣)+(𝑢+𝑣)
  • C(𝑢𝑣)+(𝑢𝑣)+(𝑢𝑣)
  • D(𝑢+𝑢)+(𝑣+𝑣)+(𝑢+𝑣)
  • E(𝑢+𝑣)+(𝑢+𝑣)+(𝑢+𝑣)

Q24:

If Aijk=4+2+3 and Bijk=2+3+4, find AB+.

  • A4+3+4ijk
  • B6+6+7ijk
  • C4+6+8ijk
  • D6+5+7ijk
  • E4+5+8ijk

Q25:

True or False: If A is any vector in 3D and 0 is the zero vector, then, for any vector A, A0+ equals 0.

  • ATrue
  • BFalse

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