Worksheet: Addition and Subtraction of Vectors

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In this worksheet, we will practice adding and subtracting vectors in 2D.

Q1:

Given that A = ⟨ 1 , 9 ⟩ and B = ⟨ − 4 , 1 ⟩ , find A B − .

  • A ⟨ 5 , 8 ⟩
  • B ⟨ − 3 , 1 0 ⟩
  • C ⟨ 1 0 , − 3 ⟩
  • D ⟨ 8 , 5 ⟩
  • E ⟨ − 8 , − 5 ⟩

Q2:

Given that A = ⟨ 9 , 5 ⟩ , B = ⟨ − 1 0 , 3 ⟩ , and C = ⟨ − 3 , 6 ⟩ , find A B C + − .

  • A ⟨ − 2 2 , 4 ⟩
  • B ⟨ − 4 , 1 4 ⟩
  • C ⟨ 2 2 , − 4 ⟩
  • D ⟨ 2 , 2 ⟩
  • E ⟨ 1 6 , 8 ⟩

Q3:

Given that A = ⟨ − 2 , 2 ⟩ , B = ⟨ 5 , 2 ⟩ , and C = ⟨ − 3 , − 2 ⟩ , find − + − A B C .

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

Q4:

Given that A = ⟨ − 1 , 2 ⟩ and B = ⟨ 3 , − 6 ⟩ , find 6 + 2 A B .

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

Q5:

A woman started walking from home and walked 6 miles at 4 0 ∘ north of east, then 2 miles at 1 5 ∘ east of south, then 5 miles at 3 0 ∘ south of west. If she walked straight home, how far would she have to walk and in what direction? Give the distance in miles correct to 2 decimal places and the direction in degrees correct to one decimal place.

  • A 1.93 miles, 5 6 . 6 ∘ west of north
  • B 6.06 miles, 8 6 . 2 ∘ north of west
  • C 0.97 miles, 3 6 . 3 ∘ north of west
  • D 6.06 miles, 8 6 . 2 ∘ west of north
  • E 1.93 miles, 2 7 . 5 ∘ east of north

Q6:

Given that B = ⟨ − 9 , − 3 ⟩ , C = ⟨ − 4 , − 2 ⟩ , and D = ⟨ − 2 , 9 ⟩ , determine the vector A that satisfies the equation A B C D = − 4 + 2 − 6 .

  • A ⟨ 4 0 , − 4 6 ⟩
  • B ⟨ − 4 0 , 4 6 ⟩
  • C ⟨ 5 6 , − 3 8 ⟩
  • D ⟨ − 5 6 , 3 8 ⟩

Q7:

Given that A = ⟨ 9 , − 1 2 ⟩ and B = ⟨ 1 3 , − 1 ⟩ , find A B + .

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

Q8:

If ‖ ‖ = 5 u and ‖ ‖ = 2 v , what is the smallest that ‖ + ‖ u v could be?

Q9:

Are there any vector u and v for which ‖ + ‖ > ‖ ‖ + ‖ ‖ u v u v ?

  • Ano
  • Byes

Q10:

Let u = ⟨ 3 , − 2 ⟩ and v = ⟨ − 9 , 5 ⟩ .

What are the components of u v + ?

  • A ⟨ 3 , − 6 ⟩
  • B ⟨ 8 , − 1 1 ⟩
  • C ⟨ 1 2 , − 7 ⟩
  • D ⟨ − 6 , 3 ⟩
  • E ⟨ − 2 7 , − 1 0 ⟩

What are the components of v u + ?

  • A ⟨ 8 , − 1 1 ⟩
  • B ⟨ 1 2 , − 7 ⟩
  • C ⟨ − 6 , 3 ⟩
  • D ⟨ 3 , − 6 ⟩
  • E ⟨ − 2 7 , − 1 0 ⟩

Q11:

Consider the vectors u = ⟨ 1 , 2 ⟩ and v = ⟨ − 3 , − 1 ⟩ .

What is the magnitude of u ? Give your answer correct to two decimal places if needed.

What is the magnitude of v ? Give your answer correct to two decimal places if needed.

What is the magnitude of u v + ? Give your answer correct to two decimal if needed.

Q12:

Consider the vectors u = ⟨ 5 , − 2 ⟩ and v = ⟨ − 5 , 2 ⟩ .

What is the magnitude of u ? Give your answer correct to two decimal places if needed.

What is the magnitude of v ? Give your answer correct to two decimal places if needed.

What is the magnitude of u v + ? Give your answer correct to two decimal places if needed.

Q13:

Given that A i j = − 5 + 1 0 and B i j = − 4 − 5 , where i and j are two perpendicular unit vectors, find A B − .

  • A 5 − 9 i j
  • B − 9 + 5 i j
  • C 1 5 − i j
  • D − + 1 5 i j
  • E − 1 5 + i j

Q14:

Given that A i j = − 5 − 6 and B i j = − 4 − 6 , where i and j are two perpendicular unit vectors, find | 2 − 2 | A B .

  • A2
  • B − 2 i
  • C4
  • D − 2
  • E − 9 − 1 2 i j

Q15:

Given that A = ⟨ − 3 , − 5 ⟩ , B A ∥ and | | = 4 √ 3 4 B , find B .

  • A ⟨ − 8 , − 1 6 ⟩
  • B ⟨ − 3 , − 5 ⟩
  • C ⟨ − 1 2 , − 2 0 ⟩
  • D ⟨ − 1 6 , − 2 4 ⟩

Q16:

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

What are the components of u ?

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

What are the components of v ?

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

What are the components of u v + ?

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

Q17:

If A = ⟨ 6 , − 4 , 7 ⟩ , and B = ⟨ 5 , 6 , 4 ⟩ , determine A B + .

  • A ⟨ 2 , 1 1 , 1 1 ⟩
  • B ⟨ 1 1 , 2 , 1 1 ⟩
  • C ⟨ − 1 , 0 , − 3 ⟩
  • D ⟨ 1 , − 1 0 , 3 ⟩

Q18:

A woman started walking from home and walked 4 miles east, 7 miles southeast, 6 miles south, 5 miles southwest, and 3 miles east. How far did she walk in total? If she walked in a straight line back home, how far would she have to walk? Give your answer correct to three decimal places if necessary.

  • A 15 miles, 16.752 miles
  • B25 miles, 16.752 miles
  • C 15 miles, 21.204 miles
  • D 9 miles, 12.727 miles
  • E 25 miles, 21.204 miles

Q19:

Fill in the blank: If v is a vector in a plane, then the vector − 2 v .

  • Amakes an angle of 𝜋 4 with v
  • Bmakes an angle of 0 with v and is twice as long
  • Cmakes an angle of 𝜋 with v and is twice as long
  • Dis perpendicular to v

Q20:

Let z be the zero vector. What is z u + equal to for any vector u ?

  • A u
  • B z

Q21:

Is it always true that 𝐴 𝐵 + 𝐶 𝐷 = 𝐴 𝐷 ?

  • Ayes
  • Bno

Q22:

Is it always true that 𝐴 𝐵 + 𝐵 𝐶 = 𝐴 𝐶 ?

  • ANo
  • BYes

Q23:

Does the vector sum ⟨ 1 , 2 ⟩ + ⟨ 2 , 3 , 1 ⟩ have a solution?

  • Ayes
  • Bno

Q24:

Let z be the zero vector. What is u z + equal to for any vector u ?

  • A u
  • B z

Q25:

Given that ( 3 , 𝑦 ) + 𝑥 ( 3 , 5 ) − ( 0 , 7 ) = 0 , find the values of 𝑥 and 𝑦 .

  • A 𝑥 = − 1 , 𝑦 = 2
  • B 𝑥 = 1 , 𝑦 = 2
  • C 𝑥 = 1 , 𝑦 = 1 2
  • D 𝑥 = − 1 , 𝑦 = 1 2

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