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=58+6 and Bijk=43+13, find AB.

  • A957ijk
  • B11420ijk
  • C11+19ijk
  • D91120ijk

Q2:

If A=8,9,9 and B=6,4,9, find 2545AB.

  • A8,25,185
  • 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
  • C10,0,15
  • D5,0,152

Q4:

If Aijk=97+8 and Bijk=83+4, find AB23.

  • A73+133ij
  • B73133ij
  • C253133+163ijk
  • D7313ij

Q5:

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

  • A3+3ik
  • Bijk2+3
  • Cij2
  • Dijk2+5
  • E3+3ij

Q6:

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

  • A1
  • B2
  • C23
  • D32
  • E22

Q7:

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

  • Ai
  • B89jk
  • Cijk8+9
  • D28+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
  • B155,155
  • C15,15
  • D15,15

Q9:

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

  • A116,116
  • B116121,116121
  • C22911,22911
  • D229,229

Q10:

Given Ajk=3+5 and Bijk=2+3+4, determine the unit vector in the direction BA.

  • A4141(2+6)ijk
  • B8585(2+9)ik
  • C721(2+6)ijk
  • D4141(26+)ijk

Q11:

Find the unit vector in the direction of 𝐴𝐵, where 𝐴4,5,2 and 𝐵9,2,3.

  • A5,7,1
  • B1,1,1
  • C13,753,153
  • D13,753,153

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+.

  • A5,0,2
  • B7
  • C5,0,2
  • D1,0,2
  • E1,6,2

Q15:

If A=2,3,4 and 𝐶=2, find 𝐶A.

  • A4,6,8
  • B4,6,8
  • C4,6,8
  • D2,6,4
  • E4,6,8

Q16:

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

  • A33
  • B3
  • C13
  • D9
  • E3

Q17:

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

  • A7,11,2
  • B2,11,1
  • C7,15,8
  • D1,15,2
  • E6,11,6

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+.

  • A6,4,5
  • B4,7,2
  • C6,7,5
  • D4,7,1
  • E4,4,2

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(22)ijk newtons
  • B3i 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+

  • A𝑢+𝑣,𝑢+𝑣,𝑢+𝑣
  • B𝑢𝑣,𝑢𝑣,𝑢𝑣
  • C𝑢𝑣,𝑢𝑣,𝑢𝑣
  • D𝑢+𝑣,𝑢+𝑣,𝑢+𝑣
  • E𝑢𝑣,𝑢𝑣,𝑢𝑣

AB+𝑘, where 𝑘 is a constant

  • A𝑘𝑢+𝑣,𝑘𝑢+𝑣,𝑘𝑢+𝑣
  • B𝑘𝑢𝑣,𝑘𝑢𝑣,𝑘𝑢𝑣
  • C𝑢+𝑘𝑣,𝑢+𝑘𝑣,𝑢+𝑘𝑣
  • D𝑢+𝑣,𝑢+𝑣,𝑢+𝑣
  • E𝑘𝑢𝑣,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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