Worksheet: Vectors in Terms of Fundamental Unit Vectors

In this worksheet, we will practice expressing vectors in terms of fundamental unit vectors instead of component form and adding and subtracting vectors in that form.

Q1:

Find the unit vector in the direction of the 𝑦 -axis.

  • A | 0 , 0 , 1 |
  • B | 1 , 0 , 0 |
  • C | 1 , 0 , 1 |
  • D | 0 , 1 , 0 |
  • E | 1 , 1 , 1 |

Q2:

Find the unit vector in the direction of the 𝑧 -axis.

  • A ( 1 , 0 , 0 )
  • B ( 0 , 1 , 0 )
  • C ( 1 , 1 , 0 )
  • D ( 0 , 0 , 1 )
  • E ( 1 , 1 , 1 )

Q3:

If 𝐴 = 4 𝑖 + 4 𝑗 5 𝑘 and 𝐵 = 3 𝑖 𝑘 , determine 𝐴 𝐵 .

  • A 3 2
  • B3
  • C 3 3

Q4:

If 𝐴 = 5 𝑖 2 𝑗 𝑘 and 𝐵 = 𝑗 + 2 𝑘 , determine 𝐴 + 𝐵 and 𝐴 + 𝐵 .

  • A 𝐴 + 𝐵 = 3 2 , 𝐴 + 𝐵 = 2 + 2
  • B 𝐴 + 𝐵 = 3 , 𝐴 + 𝐵 = 1 + 2
  • C 𝐴 + 𝐵 = 3 5 , 𝐴 + 𝐵 = 5 + 3 0

Q5:

Suppose that 𝐴 = ( 5 , 9 , 9 ) , 𝐵 = ( 4 , 2 , 9 ) , and 𝐴 + 𝐵 + 𝐶 = 𝑖 . What is 𝐶 ?

  • A 2 𝑖 + 7 𝑗
  • B 𝑖 7 𝑗
  • C 𝑖
  • D 7 𝑗

Q6:

Given that 𝐴 = 3 𝑖 + 𝑗 + 𝑚 𝑘 and that 𝐵 is a unit vector equal to 1 5 𝐴 , determine the possible values of 𝑚 .

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

Q7:

Given that 𝐴 and 𝐵 are two unit vectors, and 𝐴 + 𝐵 = 1 , evaluate 6 𝐴 + 4 𝐵 2 𝐴 + 𝐵 .

Q8:

If and are unit vectors and the measure of the angle between them, find .

  • A
  • B
  • C
  • D
  • E

Q9:

Find the unit vector in the direction of the 𝑥 -axis.

  • A ( 0 , 0 , 1 )
  • B ( 0 , 1 , 0 )
  • C ( 0 , 1 , 1 )
  • D ( 1 , 0 , 0 )
  • E ( 1 , 1 , 1 )

Q10:

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

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

Q11:

Is A = 2 3 , 2 3 , 1 4 a unit vector?

  • ANo
  • BYes

Q12:

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

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

Q13:

If 𝐴 = 3 𝑖 𝑗 4 𝑘 and 𝐵 = 3 𝑖 + 5 𝑗 5 𝑘 , determine 𝐴 𝐵 .

  • A 2 6
  • B 1 3
  • C 7 3

Q14:

If 𝐴 = 2 𝑖 3 𝑗 + 4 𝑘 and 𝐵 = 4 𝑖 4 𝑗 3 𝑘 , determine 𝐴 + 𝐵 and 𝐴 + 𝐵 .

  • A 𝐴 + 𝐵 = 2 7 , 𝐴 + 𝐵 = 2 + 2 2
  • B 𝐴 + 𝐵 = 1 4 , 𝐴 + 𝐵 = 1 + 1 1
  • C 𝐴 + 𝐵 = 8 6 , 𝐴 + 𝐵 = 2 9 + 4 1

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