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Lesson: Vectors in Terms of Fundamental Unit Vectors

Worksheet • 14 Questions

Q1:

Find the unit vector in the direction of the π‘₯ -axis.

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

Q2:

Given that ⃑ 𝐴 and ⃑ 𝐡 are two unit vectors, and β€– β€– ⃑ 𝐴 + ⃑ 𝐡 β€– β€– = 1 , evaluate ο€Ί 6 ⃑ 𝐴 + 4 ⃑ 𝐡  βŠ™ ο€Ί βˆ’ 2 ⃑ 𝐴 + ⃑ 𝐡  .

Q3:

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

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

Q4:

Suppose that ⃑ 𝐴 = ( 5 , 9 , 9 ) , ⃑ 𝐡 = ( βˆ’ 4 , βˆ’ 2 , βˆ’ 9 ) , and ⃑ 𝐴 + ⃑ 𝐡 + ⃑ 𝐢 = ⃑ 𝑖 . What is ⃑ 𝐢 ?

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

Q5:

Suppose that ⃑ 𝐴 = ( 4 , 7 , βˆ’ 7 ) , ⃑ 𝐡 = ( βˆ’ 5 , 1 , βˆ’ 2 ) , and ⃑ 𝐴 + ⃑ 𝐡 + ⃑ 𝐢 = ⃑ 𝑖 . What is ⃑ 𝐢 ?

  • A 2 ⃑ 𝑖 βˆ’ 8 ⃑ 𝑗 + 9 ⃑ π‘˜
  • B ⃑ 𝑖 βˆ’ 8 ⃑ 𝑗 + 9 ⃑ π‘˜
  • C ⃑ 𝑖
  • D 8 ⃑ 𝑗 βˆ’ 9 ⃑ π‘˜

Q6:

Given that ⃑ 𝐴 = 3 ⃑ 𝑖 + ⃑ 𝑗 + π‘š ⃑ π‘˜ and that ⃑ 𝐡 is a unit vector equal to 1 5 ⃑ 𝐴 , determine the possible values of π‘š .

  • A √ 1 5 , βˆ’ √ 1 5
  • B 1 5 , βˆ’ 1 5
  • C 3 5 , βˆ’ 3 5
  • D √ 1 5 5 , βˆ’ √ 1 5 5

Q7:

Is ⃑ 𝐴 = ο€Ό 2 3 , 2 3 , 1 4  a unit vector?

  • ATrue
  • BFalse

Q8:

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

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

Q9:

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

  • A
  • B
  • C
  • D
  • E

Q10:

Suppose a unit vector ⃑ 𝐴 is such that 1 1 ⃑ 𝐴 = ( βˆ’ 1 , βˆ’ 2 , π‘˜ ) . Determine the possible values of π‘˜ .

  • A 2 √ 2 9 , βˆ’ 2 √ 2 9
  • B 1 1 6 , βˆ’ 1 1 6
  • C 1 1 6 1 2 1 , βˆ’ 1 1 6 1 2 1
  • D 2 √ 2 9 1 1 , βˆ’ 2 √ 2 9 1 1

Q11:

If ⃑ 𝐴 = 4 ⃑ 𝑖 + 4 ⃑ 𝑗 βˆ’ 5 ⃑ π‘˜ and ⃑ 𝐡 = 3 ⃑ 𝑖 βˆ’ ⃑ π‘˜ , determine β€– β€– ⃑ 𝐴 βˆ’ ⃑ 𝐡 β€– β€– .

  • A3
  • B √ 3 3
  • C 3 √ 2

Q12:

If ⃑ 𝐴 = 3 ⃑ 𝑖 βˆ’ ⃑ 𝑗 βˆ’ 4 ⃑ π‘˜ and ⃑ 𝐡 = βˆ’ 3 ⃑ 𝑖 + 5 ⃑ 𝑗 βˆ’ 5 ⃑ π‘˜ , determine β€– β€– ⃑ 𝐴 βˆ’ ⃑ 𝐡 β€– β€– .

  • A √ 1 3
  • B √ 7 3
  • C √ 2 6

Q13:

If ⃑ 𝐴 = 5 ⃑ 𝑖 βˆ’ 2 ⃑ 𝑗 βˆ’ ⃑ π‘˜ and ⃑ 𝐡 = βˆ’ ⃑ 𝑗 + 2 ⃑ π‘˜ , determine β€– β€– ⃑ 𝐴 + ⃑ 𝐡 β€– β€– and β€– β€– ⃑ 𝐴 β€– β€– + β€– β€– ⃑ 𝐡 β€– β€– .

  • A β€– β€– ⃑ 𝐴 + ⃑ 𝐡 β€– β€– = 3 , β€– β€– ⃑ 𝐴 β€– β€– + β€– β€– ⃑ 𝐡 β€– β€– = 1 + √ 2
  • B β€– β€– ⃑ 𝐴 + ⃑ 𝐡 β€– β€– = √ 3 5 , β€– β€– ⃑ 𝐴 β€– β€– + β€– β€– ⃑ 𝐡 β€– β€– = √ 5 + √ 3 0
  • C β€– β€– ⃑ 𝐴 + ⃑ 𝐡 β€– β€– = 3 √ 2 , β€– β€– ⃑ 𝐴 β€– β€– + β€– β€– ⃑ 𝐡 β€– β€– = √ 2 + 2

Q14:

If ⃑ 𝐴 = βˆ’ 2 ⃑ 𝑖 βˆ’ 3 ⃑ 𝑗 + 4 ⃑ π‘˜ and ⃑ 𝐡 = βˆ’ 4 ⃑ 𝑖 βˆ’ 4 ⃑ 𝑗 βˆ’ 3 ⃑ π‘˜ , determine β€– β€– ⃑ 𝐴 + ⃑ 𝐡 β€– β€– and β€– β€– ⃑ 𝐴 β€– β€– + β€– β€– ⃑ 𝐡 β€– β€– .

  • A β€– β€– ⃑ 𝐴 + ⃑ 𝐡 β€– β€– = √ 1 4 , β€– β€– ⃑ 𝐴 β€– β€– + β€– β€– ⃑ 𝐡 β€– β€– = 1 + √ 1 1
  • B β€– β€– ⃑ 𝐴 + ⃑ 𝐡 β€– β€– = √ 8 6 , β€– β€– ⃑ 𝐴 β€– β€– + β€– β€– ⃑ 𝐡 β€– β€– = √ 2 9 + √ 4 1
  • C β€– β€– ⃑ 𝐴 + ⃑ 𝐡 β€– β€– = 2 √ 7 , β€– β€– ⃑ 𝐴 β€– β€– + β€– β€– ⃑ 𝐡 β€– β€– = √ 2 + √ 2 2
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