Worksheet: Vector Operations

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In this worksheet, we will practice solving operations on vectors algebraically such as vector addition, vector subtraction, and scalar multiplication.

Q1:

Given that ⃑ 𝐴 = ( 0 , 1 ) and ⃑ 𝐵 = ( − 3 , − 6 ) , find 3 2 ⃑ 𝐴 − ⃑ 𝐵 .

  • A 9 2 , − 1 5 2
  • B − 9 2 , − 1 5 2
  • C − 9 2 , 2 1 2
  • D 9 2 , 2 1 2

Q2:

Given that ⃑ 𝐴 = ( 0 , − 1 ) and ‖ ‖ 𝑘 ⃑ 𝐴 ‖ ‖ = 1 2 , find the possible values of 𝑘 .

  • A 1 1 2 , − 1 1 2
  • B12
  • C 1 1 2
  • D 1 2 , − 1 2

Q3:

Given that ⃑ 𝐴 = ( 2 , − 4 ) and ⃑ 𝐵 = ( − 7 , − 6 ) , find ⃑ 𝐴 − 4 ⃑ 𝐵 .

  • A ( − 2 6 , 2 0 )
  • B ( − 2 6 , − 2 8 )
  • C ( 3 0 , − 2 8 )
  • D ( 3 0 , 2 0 )

Q4:

A force of newtons is being applied to an object. What other force should be applied to achieve a total force of newtons?

  • A newtons
  • B newtons
  • C newtons
  • D newtons
  • E newtons

Q5:

Given that u = ⟨ 0 , 4 ⟩ and v = ⟨ 0 , − 5 ⟩ , find the components of u v + .

  • A ⟨ 0 , − 2 0 ⟩
  • B ⟨ 0 , 9 ⟩
  • C ⟨ − 5 , 4 ⟩
  • D ⟨ 0 , − 1 ⟩
  • E ⟨ 0 , 1 ⟩

Q6:

Given that u = ⟨ 2 , − 3 ⟩ , v = ⟨ − 5 , 4 ⟩ , and w = ⟨ 3 , − 1 ⟩ , find the components of u v w + + .

  • A ⟨ − 1 5 , 1 2 ⟩
  • B ⟨ − 3 , 1 ⟩
  • C ⟨ 4 , − 6 ⟩
  • D ⟨ 0 , 0 ⟩
  • E ⟨ 0 , − 2 ⟩

Q7:

Given that u = ⟨ 2 , − 3 ⟩ , v = ⟨ 3 , 2 ⟩ , and w = ⟨ − 1 , − 5 ⟩ , find the components of u v w + + .

  • A ⟨ − 6 , 3 0 ⟩
  • B ⟨ 5 , − 1 ⟩
  • C ⟨ 0 , 0 ⟩
  • D ⟨ 4 , − 6 ⟩
  • E ⟨ − 6 , 4 ⟩

Q8:

Given that u = ⟨ 2 , − 4 ⟩ and v = ⟨ 0 , 0 ⟩ , find the components of u v + .

  • A ⟨ − 2 , 4 ⟩
  • B ⟨ 0 , 0 ⟩
  • C ⟨ − 4 , 2 ⟩
  • D ⟨ 2 , − 4 ⟩
  • E ⟨ 4 , − 2 ⟩

Q9:

Shown on the grid of unit squares are the vectors u , v , and u v + .

What are the components of u ?

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

What are the components of v ?

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

What are the components of u v + ?

  • A ⟨ − 2 , 3 ⟩
  • B ⟨ 1 , 2 ⟩
  • C ⟨ − 1 , 2 ⟩
  • D ⟨ − 1 , 3 ⟩
  • E ⟨ 1 , 3 ⟩

Q10:

The figure shows a regular hexagon 𝐴 𝐵 𝐶 𝐷 𝐸 𝐹 divided into 6 equilateral triangles. Which of the following is equal to 𝐵 𝐸 + 𝐹 𝐴 ?

  • A 𝐵 𝐴
  • B 𝐷 𝐶
  • C 𝐸 𝐹
  • D 𝐶 𝐷
  • E 𝐹 𝐷

Q11:

Given that u = ⟨ 2 , − 4 ⟩ and v = ⟨ − 2 , 4 ⟩ , find the components of u v + .

  • A ⟨ 4 , 8 ⟩
  • B ⟨ − 4 , − 8 ⟩
  • C ⟨ 0 , − 8 ⟩
  • D ⟨ 0 , 0 ⟩
  • E ⟨ − 4 , − 1 6 ⟩

Q12:

Given that u = ⟨ − 3 , − 1 ⟩ , and v = ⟨ − 2 , 5 ⟩ , find the components of u v + .

  • A ⟨ − 1 , − 6 ⟩
  • B ⟨ 6 , − 5 ⟩
  • C ⟨ 1 , 6 ⟩
  • D ⟨ − 5 , 4 ⟩
  • E ⟨ 2 , − 3 ⟩

Q13:

Given that u = ⟨ 3 , 1 ⟩ and v = ⟨ 2 , 5 ⟩ find the components of u + v .

  • A ⟨ 8 , 3 ⟩
  • B ⟨ 6 , 5 ⟩
  • C ⟨ 1 , − 4 ⟩
  • D ⟨ 5 , 6 ⟩
  • E ⟨ − 1 , 4 ⟩

Q14:

and Find .

Q15:

On a lattice, where 𝐴 𝐶 = ( − 5 , − 5 ) , 𝐵 𝐶 = ( − 1 2 , 6 ) , and 3 ⃑ 𝐶 + 𝐴 𝐵 = ( − 8 , 1 3 ) , determine the coordinates of the point 𝐵 .

  • A ( − 1 7 , 1 4 )
  • B ( − 1 0 , 3 )
  • C ( − 1 , 6 )
  • D ( 7 , 2 )

Q16:

On a lattice, where 𝐴 𝐶 = ( 3 , 3 ) , 𝐵 𝐶 = ( 1 3 , − 7 ) , and 2 ⃑ 𝐶 + 2 𝐴 𝐵 = ( − 4 , − 4 ) , find the coordinates of the point 𝐶 .

  • A ( − 1 8 , 2 )
  • B ( − 1 2 , 8 )
  • C ( 1 4 , − 6 )
  • D ( 8 , − 1 2 )
  • E ( 1 6 , − 2 4 )

Q17:

Given that ⃑ 𝐴 = ( − 4 , − 1 ) and ⃑ 𝐵 = ( − 2 , − 1 ) , express ⃑ 𝐶 = ( − 8 , − 1 ) in terms of ⃑ 𝐴 and ⃑ 𝐵 .

  • A 7 ⃑ 𝐴 + 1 0 ⃑ 𝐵
  • B − ⃑ 𝐴 + 6 ⃑ 𝐵
  • C 5 ⃑ 𝐴 − 6 ⃑ 𝐵
  • D 3 ⃑ 𝐴 − 2 ⃑ 𝐵

Q18:

If ⃑ 𝐴 = ( 1 , 2 , 1 ) , ⃑ 𝐵 = ( − 1 , − 1 , 0 ) , and ⃑ 𝐶 = ( − 2 , − 1 , 1 ) , express ⃑ 𝐶 in terms of ⃑ 𝐴 and ⃑ 𝐵 .

  • A ⃑ 𝐶 = − 2 ⃑ 𝐴 − ⃑ 𝐵
  • B ⃑ 𝐶 = 3 ⃑ 𝐴 + ⃑ 𝐵
  • C ⃑ 𝐶 = ⃑ 𝐴 + ⃑ 𝐵
  • D ⃑ 𝐶 = ⃑ 𝐴 + 3 ⃑ 𝐵

Q19:

When is it true that u v v u + = + ?

  • Aonly when u and v are parallel
  • Bonly when u and v are equivalent
  • Conly when u and v are perpendicular
  • Dfor any vectors u and v
  • Eonly when u and v are not perpendicular

Q20:

When is it true that ‖ + ‖ = ‖ ‖ + ‖ ‖ u v u v ?

  • Anever
  • Balways
  • Cwhen u and v are equivalent vectors
  • Dwhen u and v are parallel vectors
  • Ewhen u and v are perpendicular vectors

Q21:

⃑ 𝑉 and 𝑊 are two vectors, where ⃑ 𝑉 = ( − 1 , 5 , − 2 ) and 𝑊 = ( 3 , 1 , 1 ) . Comparing ‖ ‖ ⃑ 𝑉 + 𝑊 ‖ ‖ and ‖ ‖ ⃑ 𝑉 ‖ ‖ + ‖ ‖ 𝑊 ‖ ‖ , which quantity is larger?

  • AThey are both the same.
  • B ‖ ‖ ⃑ 𝑉 + 𝑊 ‖ ‖
  • C ‖ ‖ ⃑ 𝑉 ‖ ‖ + ‖ ‖ 𝑊 ‖ ‖

Q22:

Find all the possible values of 𝑚 given ⃑ 𝐴 = ( − 4 , 3 , 1 ) , ⃑ 𝐵 = ( 6 , − 6 , 𝑚 − 1 3 ) and ‖ ‖ ⃑ 𝐴 + ⃑ 𝐵 ‖ ‖ = 7 .

  • A − 5
  • B1
  • C − 1 8 , − 6
  • D18, 6

Q23:

Given that ⃑ 𝐴 = ( 3 , 3 ) and ⃑ 𝐵 = ( 4 , 6 ) , find 1 2 ⃑ 𝐴 + ⃑ 𝐵 .

  • A 7 2 , − 3 2
  • B − 1 2 , − 3 2
  • C − 1 2 , 9 2
  • D 7 2 , 9 2

Q24:

Given that ⃑ 𝐴 = ( − 9 , 4 ) and ⃑ 𝐵 = ( 3 , − 2 ) , find − 1 2 ⃑ 𝐴 + ⃑ 𝐵 .

  • A ( 3 , − 3 )
  • B ( 6 , − 3 )
  • C ( 6 , − 1 )
  • D ( 3 , − 1 )

Q25:

Given that ⃑ 𝐴 = ( − 7 , 6 ) and ⃑ 𝐵 = ( 9 , − 7 ) , find 3 2 ⃑ 𝐴 − ⃑ 𝐵 .

  • A − 2 4 , − 3 2
  • B 3 , − 3 2
  • C 3 , 3 9 2
  • D − 2 4 , 3 9 2

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