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Worksheet: Adding and Subtracting Vectors

Q1:

Given that ⃑ 𝐴 = ( 1 , 9 ) and ⃑ 𝐡 = ( βˆ’ 4 , 1 ) , find ⃑ 𝐴 βˆ’ ⃑ 𝐡 .

  • A ( 1 0 , βˆ’ 3 )
  • B ( βˆ’ 3 , 1 0 )
  • C ( βˆ’ 8 , βˆ’ 5 )
  • D ( 5 , 8 )
  • E ( 8 , 5 )

Q2:

Given that ⃑ 𝐴 = ( 9 , 5 ) , ⃑ 𝐡 = ( βˆ’ 1 0 , 3 ) , and ⃑ 𝐢 = ( βˆ’ 3 , 6 ) , find ⃑ 𝐴 + ⃑ 𝐡 βˆ’ ⃑ 𝐢 .

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

Q3:

Given that ⃑ 𝐴 = ( βˆ’ 2 , 2 ) , ⃑ 𝐡 = ( 5 , 2 ) , and ⃑ 𝐢 = ( βˆ’ 3 , βˆ’ 2 ) , find βˆ’ ⃑ 𝐴 + ⃑ 𝐡 βˆ’ ⃑ 𝐢 .

  • A ( 4 , βˆ’ 2 )
  • B ( 0 , βˆ’ 2 )
  • C ( 0 , 2 )
  • D ( 1 0 , 2 )

Q4:

Given that ⃑ 𝐴 = ( βˆ’ 6 , 3 ) and ⃑ 𝐡 = ( 8 , 7 ) , find ⃑ 𝐴 + ⃑ 𝐡 .

  • A ( βˆ’ 1 4 , 1 0 )
  • B ( 2 , βˆ’ 4 )
  • C ( 2 , 4 )
  • D ( 2 , 1 0 )
  • E ( 1 4 , 1 0 )

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

Q6:

Given that ⃑ 𝐡 = ( βˆ’ 9 , βˆ’ 3 ) , ⃑ 𝐢 = ( βˆ’ 4 , βˆ’ 2 ) , and ⃑ 𝐷 = ( βˆ’ 2 , 9 ) , determine the vector ⃑ 𝐴 that satisfies the equation ⃑ 𝐴 = βˆ’ 4 ⃑ 𝐡 + 2 ⃑ 𝐢 βˆ’ 6 ⃑ 𝐷 .

  • A ( βˆ’ 4 0 , 4 6 )
  • B ( 5 6 , βˆ’ 3 8 )
  • C ( βˆ’ 5 6 , 3 8 )
  • D ( 4 0 , βˆ’ 4 6 )

Q7:

Given that ⃑ 𝐴 = ( 9 , βˆ’ 1 2 ) and ⃑ 𝐡 = ( 1 3 , βˆ’ 1 ) , find ⃑ 𝐴 + ⃑ 𝐡 .

  • A ( βˆ’ 2 2 , 1 3 )
  • B ( βˆ’ 1 3 , 2 2 )
  • C ( 1 3 , βˆ’ 2 2 )
  • D ( 2 2 , βˆ’ 1 3 )

Q8:

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

Q9:

Are there any vector u and v for which β€– + β€– > β€– β€– + β€– β€– u v u v ?

  • Ano
  • Byes

Q10:

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

What are the components of u v + ?

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

What are the components of v u + ?

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

Q11:

Consider the vectors u = ⟨ 1 , 2 ⟩ and v = ⟨ βˆ’ 3 , βˆ’ 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 , βˆ’ 2 ⟩ and v = ⟨ βˆ’ 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 = βˆ’ 5 ⃑ 𝑖 + 1 0 ⃑ 𝑗 and ⃑ 𝐡 = βˆ’ 4 ⃑ 𝑖 βˆ’ 5 ⃑ 𝑗 , where ⃑ 𝑖 and ⃑ 𝑗 are two perpendicular unit vectors, find ⃑ 𝐴 βˆ’ ⃑ 𝐡 .

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

Q14:

Given that ⃑ 𝐴 = βˆ’ 5 ⃑ 𝑖 βˆ’ 6 ⃑ 𝑗 and ⃑ 𝐡 = βˆ’ 4 ⃑ 𝑖 βˆ’ 6 ⃑ 𝑗 , where ⃑ 𝑖 and ⃑ 𝑗 are two perpendicular unit vectors, find β€– β€– 2 ⃑ 𝐴 βˆ’ 2 ⃑ 𝐡 β€– β€– .

  • A4
  • B βˆ’ 2 ⃑ 𝑖
  • C βˆ’ 2
  • D2
  • E βˆ’ 9 ⃑ 𝑖 βˆ’ 1 2 ⃑ 𝑗

Q15:

Given that A = ⟨ βˆ’ 3 , βˆ’ 5 ⟩ , B A βˆ₯ and | | = 4 √ 3 4 B , find B .

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

Q16:

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

What are the components of u ?

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

What are the components of v ?

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

What are the components of u v + ?

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

Q17:

If ⃑ 𝐴 = ( 6 , βˆ’ 4 , 7 ) , and ⃑ 𝐡 = ( 5 , 6 , 4 ) , determine ⃑ 𝐴 + ⃑ 𝐡 .

  • A ( 2 , 1 1 , 1 1 )
  • B ( 1 , βˆ’ 1 0 , 3 )
  • C ( βˆ’ 1 , 0 , βˆ’ 3 )
  • D ( 1 1 , 2 , 1 1 )

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 9 miles, 12.727 miles
  • B 15 miles, 21.204 miles
  • C 15 miles, 16.752 miles
  • D25 miles, 16.752 miles
  • E 25 miles, 21.204 miles

Q19:

Complete the sentence: If is a vector in a plane, then the vector .

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

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 𝐡 +  𝐢 𝐷 =  𝐴 𝐷 ?

  • Ano
  • Byes

Q22:

Is it always true that  𝐴 𝐡 + οƒͺ 𝐡 𝐢 =  𝐴 𝐢 ?

  • AYes
  • BNo

Q23:

Does the vector sum ( 1 , 2 ) + ( 2 , 3 , 1 ) have a solution?

  • Ano
  • Byes

Q24:

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

  • A u
  • B z

Q25:

Given that , find the values of and .

  • A ,
  • B ,
  • C ,
  • D ,