Worksheet: Adding and Subtracting Vectors in 2D

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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 AB−.

  • A⟨−8,−5⟩
  • B⟨10,−3⟩
  • C⟨5,8⟩
  • D⟨8,5⟩
  • E⟨−3,10⟩

Q2:

Given that A=⟨9,5⟩, B=⟨−10,3⟩, and C=⟨−3,6⟩, find ABC+−.

  • A⟨−22,4⟩
  • B⟨2,2⟩
  • C⟨22,−4⟩
  • D⟨16,8⟩
  • E⟨−4,14⟩

Q3:

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

  • A⟨4,−2⟩
  • B⟨10,2⟩
  • C⟨0,2⟩
  • D⟨0,−2⟩

Q4:

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

  • A⟨0,24⟩
  • B⟨−12,0⟩
  • C⟨0,0⟩
  • D⟨12,0⟩
  • E⟨0,−24⟩

Q5:

A woman started walking from home and walked 6 miles at 40∘ north of east, then 2 miles at 15∘ east of south, then 5 miles at 30∘ 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.

  • A1.93 miles, 27.5∘ east of north
  • B6.06 miles, 86.2∘ north of west
  • C0.97 miles, 36.3∘ north of west
  • D1.93 miles, 56.6∘ west of north
  • E6.06 miles, 86.2∘ west of north

Q6:

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

  • A⟨56,−38⟩
  • B⟨−56,38⟩
  • C⟨−40,46⟩
  • D⟨40,−46⟩

Q7:

Given that A=⟨9,−12⟩ and B=⟨13,−1⟩, find AB+.

  • A⟨−22,13⟩
  • B⟨13,−22⟩
  • C⟨22,−13⟩
  • D⟨−13,22⟩

Q8:

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

Q9:

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

  • Ano
  • Byes

Q10:

Let u=⟨3,−2⟩ and v=⟨−9,5⟩.

What are the components of uv+?

  • A⟨3,−6⟩
  • B⟨8,−11⟩
  • C⟨12,−7⟩
  • D⟨−6,3⟩
  • E⟨−27,−10⟩

What are the components of vu+?

  • A⟨12,−7⟩
  • B⟨8,−11⟩
  • C⟨3,−6⟩
  • D⟨−6,3⟩
  • E⟨−27,−10⟩

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 uv+? 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 uv+? Give your answer correct to two decimal places if needed.

Q13:

Given that Aij=−5+10 and Bij=−4−5, where i and j are two perpendicular unit vectors, find AB−.

  • A−15+ij
  • B15−ij
  • C−9+5ij
  • D−+15ij
  • E5−9ij

Q14:

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

  • A−9−12ij
  • B2
  • C4
  • D−2i
  • E−2

Q15:

Given that A=⟨−3,−5⟩, BA∥ and ||=4√34B, find B.

  • A⟨−8,−16⟩
  • B⟨−16,−24⟩
  • C⟨−3,−5⟩
  • D⟨−12,−20⟩

Q16:

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

What are the components of u?

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

What are the components of v?

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

What are the components of uv+?

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

Q17:

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

  • A⟨1,−10,3⟩
  • B⟨11,2,11⟩
  • C⟨−1,0,−3⟩
  • D⟨2,11,11⟩

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.

  • A25 miles, 16.752 miles
  • B15 miles, 16.752 miles
  • C25 miles, 21.204 miles
  • D15 miles, 21.204 miles
  • E9 miles, 12.727 miles

Q19:

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

  • Ais perpendicular to v
  • Bmakes an angle of 𝜋 with v and is twice as long
  • Cmakes an angle of 𝜋4 with v
  • Dmakes an angle of 0 with v and is twice as long

Q20:

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

  • Az
  • Bu

Q21:

Is it always true that 𝐴𝐵+𝐶𝐷=𝐴𝐷?

  • Ayes
  • Bno

Q22:

Is it always true that 𝐴𝐵+𝐵𝐶=𝐴𝐶?

  • ANo
  • BYes

Q23:

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

  • Ano
  • Byes

Q24:

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

  • Az
  • Bu

Q25:

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

  • A𝑥=−1, 𝑦=2
  • B𝑥=1, 𝑦=2
  • C𝑥=−1, 𝑦=12
  • D𝑥=1, 𝑦=12

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