Worksheet: Vector Operations in 2D

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

Q1:

Given that A=⟨0,1⟩ and B=⟨−3,−6⟩, find 32(−)AB.

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

Q2:

Given that A=⟨0,−1⟩ and |𝑘|=12A, find the possible values of 𝑘.

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

Q3:

Given that A=⟨2,−4⟩ and B=⟨−7,−6⟩, find AB−4.

  • A ⟨ − 2 6 , − 2 8 ⟩
  • B ⟨ − 2 6 , 2 0 ⟩
  • C ⟨ 3 0 , − 2 8 ⟩
  • D ⟨ 3 0 , 2 0 ⟩

Q4:

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 ) i j k newtons
  • B − 3 i newtons
  • C ( + 2 + 2 ) i j k newtons
  • D ( 3 + + ) i j k newtons
  • E 3 i newtons

Q5:

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

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

Q6:

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

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

Q7:

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

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

Q8:

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

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

Q9:

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

What are the components of u?

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

What are the components of v?

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

What are the components of uv+?

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

Q10:

The figure shows a regular hexagon 𝐴𝐵𝐶𝐷𝐸𝐹 divided into 6 equilateral triangles. Which of the following is equal to 𝐵𝐸+𝐹𝐴?

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

Q11:

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

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

Q12:

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

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

Q13:

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

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

Q14:

A = 3 4 and B=53. Find (+)⋅ABA.

Q15:

On a lattice, where 𝐴𝐶=⟨−5,−5⟩, 𝐵𝐶=⟨−12,6⟩, and 3+𝐴𝐵=⟨−8,13⟩C, determine the coordinates of the point 𝐵.

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

Q16:

On a lattice, where 𝐴𝐶=⟨3,3⟩, 𝐵𝐶=⟨13,−7⟩, and 2+2𝐴𝐵=⟨−4,−4⟩C, find the coordinates of the point 𝐶.

  • A ( 8 , − 1 2 )
  • B ( − 1 2 , 8 )
  • C ( 1 4 , − 6 )
  • D ( − 1 8 , 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 + 1 0 A B
  • B − + 6 A B
  • C 5 − 6 A B
  • D 3 − 2 A 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 A B = +
  • B C A B = 3 +
  • C C A B = + 3
  • D C A B = − 2 −

Q19:

When is it true that uvvu+=+?

  • Afor any vectors u and v
  • Bonly when u and v are not perpendicular
  • Conly when u and v are equivalent
  • Donly when u and v are perpendicular
  • Eonly when u and v are parallel

Q20:

When is it true that ‖+‖=‖‖+‖‖uvuv?

  • Aalways
  • Bwhen u and v are perpendicular vectors
  • Cnever
  • Dwhen u and v are equivalent vectors
  • Ewhen u and v are parallel vectors

Q21:

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 | + | V W
  • B | | + | | V W
  • CThey are both the same.

Q22:

Find all the possible values of 𝑚 given A=⟨−4,3,1⟩, B=⟨6,−6,𝑚−13⟩ and |+|=7AB.

  • A − 1 8 , − 6
  • B − 5
  • C18, 6
  • D1

Q23:

Given that A=⟨8,3⟩ and B=⟨−5,3⟩, find 12(−)AB.

  • A 3 2 , 3
  • B 1 3 2 , 0
  • C 3 2 , 0
  • D 1 3 2 , 3

Q24:

Given that A=⟨−4,8⟩ and B=⟨−9,−3⟩, find 12(+)AB.

  • A 5 2 , 1 1 2
  • B − 1 3 2 , 5 2
  • C 5 2 , 5 2
  • D − 1 3 2 , 1 1 2

Q25:

Given that A=⟨−7,6⟩ and B=⟨9,−7⟩, find 32(−)AB.

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

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