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Worksheet: Component Form of a Vector

In this worksheet, we will practice expressing a vector in component form given its initial and terminal points.

Q1:

Find the components of the vector οƒ  𝐴 𝐡 , where 𝐴 = ( βˆ’ 4 . 5 , 1 . 2 ) and 𝐡 = ( βˆ’ 4 . 5 , 5 . 7 ) .

  • A ⟨ βˆ’ 4 . 5 , 1 . 2 ⟩
  • B ⟨ 4 . 5 , 0 ⟩
  • C ⟨ βˆ’ 4 . 5 , 5 . 7 ⟩
  • D ⟨ 0 , 4 . 5 ⟩
  • E ⟨ 0 , βˆ’ 4 . 5 ⟩

Q2:

Find the components of the vector οƒ  𝐴 𝐡 .

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

Q3:

The points 𝐴 , 𝐡 , and 𝐢 have coordinates ( βˆ’ 7 , 1 ) , ( βˆ’ 2 , 4 ) , and ( βˆ’ 4 , βˆ’ 1 ) respectively. Given that οƒ  𝐴 𝐡 and οƒ  𝐢 𝐷 are equivalent vectors, find the coordinates of 𝐷 .

  • A ( 9 , 4 )
  • B ( 2 , 1 )
  • C ( 4 , 9 )
  • D ( 1 , 2 )
  • E ( βˆ’ 9 , βˆ’ 4 )

Q4:

The initial point of the vector shown in the diagram is the origin, ( 0 , 0 ) .

What are the coordinates of its terminal point?

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

What are the components of the vector?

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

Q5:

What is the initial point of the vector u in the given diagram?

  • A 𝐴
  • B 𝐡

Q6:

Find the components of the vector οƒ  𝐴 𝐡 , where 𝐴 = ( 1 . 5 , βˆ’ 0 . 3 ) and 𝐡 = ( βˆ’ 1 , 5 ) .

  • A ⟨ 1 . 5 , βˆ’ 0 . 3 ⟩
  • B ⟨ 5 . 3 , βˆ’ 2 . 5 ⟩
  • C ⟨ βˆ’ 1 , 5 ⟩
  • D ⟨ βˆ’ 2 . 5 , 5 . 3 ⟩
  • E ⟨ 2 . 5 , βˆ’ 5 . 3 ⟩

Q7:

The initial point of the vector shown in the diagram is the origin, ( 0 , 0 ) .

What are the coordinates of its terminal point?

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

What are the components of the vector?

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

Q8:

Consider the vector in the given diagram.

What are the coordinates of its terminal point?

  • A ( 1 , 3 )
  • B ( βˆ’ 2 , βˆ’ 2 )
  • C ( 2 , 2 )
  • D ( 3 , 1 )
  • E ( 5 , 3 )

What are the coordinates of its initial point?

  • A ( βˆ’ 2 , βˆ’ 2 )
  • B ( 2 , 2 )
  • C ( 1 , 3 )
  • D ( 3 , 1 )
  • E ( 5 , 3 )

What are the components of the vector?

  • A ⟨ βˆ’ 5 , βˆ’ 3 ⟩
  • B ⟨ 3 , 5 ⟩
  • C ⟨ 5 , 3 ⟩
  • D ⟨ 3 , 1 ⟩
  • E ⟨ βˆ’ 2 , βˆ’ 2 ⟩

Q9:

Given the points 𝐴 ( 8 , βˆ’ 1 0 ) , 𝐡 ( 3 , 3 ) , and 𝐢 ( 9 , 1 1 ) , find the coordinates of point 𝐷 such that οƒ  𝐢 𝐷 is equivalent to οƒ  𝐴 𝐡 .

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

Q10:

Find the components of the vector οƒ  𝐴 𝐡 , where 𝐴 = ( βˆ’ 1 9 , βˆ’ 4 . 1 ) and 𝐡 = ( βˆ’ 4 . 4 , βˆ’ 9 . 7 ) .

  • A ⟨ βˆ’ 1 9 , βˆ’ 4 . 1 ⟩
  • B ⟨ βˆ’ 5 . 6 , 1 4 . 6 ⟩
  • C ⟨ βˆ’ 4 . 4 , βˆ’ 9 . 7 ⟩
  • D ⟨ 1 4 . 6 , βˆ’ 5 . 6 ⟩
  • E ⟨ βˆ’ 1 4 . 6 , 5 . 6 ⟩

Q11:

What is the magnitude of the vector οƒ  𝐴 𝐡 , where 𝐴 = ( 9 . 1 , βˆ’ 2 . 3 ) and 𝐡 = ( 7 . 7 , 2 . 5 ) ?

Q12:

The points 𝐴 , 𝐡 , and 𝐢 have coordinates ( 2 , 5 ) , ( 3 , 7 ) , and ( 5 , 9 ) respectively. Given that οƒ  𝐴 𝐡 and οƒ  𝐢 𝐷 are equivalent vectors, find the coordinates of 𝐷 .

  • A ( 4 , 7 )
  • B ( 1 1 , 6 )
  • C ( 7 , 4 )
  • D ( 6 , 1 1 )
  • E ( βˆ’ 4 , βˆ’ 7 )

Q13:

A vector has components ⟨ βˆ’ 1 , βˆ’ 2 ⟩ and a terminal point ( βˆ’ 9 , 0 ) . What is its initial point?

  • A ( βˆ’ 1 0 , βˆ’ 2 )
  • B ( 8 , βˆ’ 2 )
  • C ( 2 , βˆ’ 8 )
  • D ( βˆ’ 8 , 2 )
  • E ( βˆ’ 8 , βˆ’ 2 )

Q14:

Find the coordinates of the initial point of the vector in the given diagram.

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

Q15:

Consider the vector in the given diagram.

What are the coordinates of its terminal point?

  • A ( βˆ’ 2 , βˆ’ 4 )
  • B ( βˆ’ 4 , 3 )
  • C ( 3 , βˆ’ 4 )
  • D ( βˆ’ 4 , βˆ’ 2 )
  • E ( 0 , βˆ’ 5 )

What are the coordinates of its initial point?

  • A ( βˆ’ 4 , 3 )
  • B ( 3 , βˆ’ 4 )
  • C ( βˆ’ 2 , βˆ’ 4 )
  • D ( βˆ’ 4 , βˆ’ 2 )
  • E ( 0 , βˆ’ 5 )

What are the components of the vector?

  • A ⟨ 0 , 5 ⟩
  • B ⟨ βˆ’ 5 , 0 ⟩
  • C ⟨ 0 , βˆ’ 5 ⟩
  • D ⟨ βˆ’ 4 , 3 ⟩
  • E ⟨ βˆ’ 4 , βˆ’ 2 ⟩

Q16:

Find the components of the vector οƒ  𝐴 𝐡 , where 𝐴 = ( 1 5 . 3 , βˆ’ 5 ) and 𝐡 = ( βˆ’ 3 . 7 , βˆ’ 5 ) .

  • A ⟨ 1 5 . 3 , βˆ’ 5 ⟩
  • B ⟨ 0 , βˆ’ 1 9 ⟩
  • C ⟨ βˆ’ 3 . 7 , βˆ’ 5 ⟩
  • D ⟨ βˆ’ 1 9 , 0 ⟩
  • E ⟨ 1 9 , 0 ⟩

Q17:

What is the terminal point of the vector u in the given diagram?

  • A 𝐡
  • B 𝐴

Q18:

The initial point of the vector shown in the diagram is the origin, ( 0 , 0 ) .

What are the coordinates of its terminal point?

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

What are the components of the vector?

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

Q19:

Consider the vector in the given diagram.

What are the coordinates of its terminal point?

  • A ( 1 , 2 )
  • B ( 3 , 4 )
  • C ( 4 , 3 )
  • D ( 2 , 1 )
  • E ( βˆ’ 1 , βˆ’ 3 )

What are the coordinates of its initial point?

  • A ( 3 , 4 )
  • B ( 4 , 3 )
  • C ( 1 , 2 )
  • D ( 2 , 1 )
  • E ( βˆ’ 1 , βˆ’ 3 )

What are the components of the vector?

  • A ⟨ 1 , 3 ⟩
  • B ⟨ βˆ’ 3 , βˆ’ 1 ⟩
  • C ⟨ βˆ’ 1 , βˆ’ 3 ⟩
  • D ⟨ 2 , 1 ⟩
  • E ⟨ 3 , 4 ⟩

Q20:

Using the shown graph, determine the position vector of point relative to the origin point , denoted by , and find its norm, denoted by .

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

Q21:

Consider the vector in the given diagram.

What are the coordinates of its terminal point?

  • A ( 4 , 1 )
  • B ( 7 , 4 )
  • C ( 7 , 1 )
  • D ( 1 , 4 )
  • E ( βˆ’ 6 , 0 )

What are the coordinates of its initial point?

  • A ( 7 , 4 )
  • B ( 4 , 1 )
  • C ( 4 , 7 )
  • D ( 1 , 4 )
  • E ( βˆ’ 6 , 0 )

What are the components of the vector?

  • A ⟨ 6 , 0 ⟩
  • B ⟨ 0 , βˆ’ 6 ⟩
  • C ⟨ βˆ’ 6 , 0 ⟩
  • D ⟨ 7 , 4 ⟩
  • E ⟨ 1 , 4 ⟩

Q22:

Find the coordinates of the terminal point of the vector in the given diagram.

  • A ( βˆ’ 3 , βˆ’ 2 )
  • B ( 1 , 4 )
  • C ( βˆ’ 4 , βˆ’ 1 )
  • D ( 4 , 1 )
  • E ( βˆ’ 2 , βˆ’ 3 )

Q23:

Find the components of the vector οƒ  𝐴 𝐡 .

  • A ⟨ 4 , βˆ’ 1 . 7 ⟩
  • B ⟨ 1 . 9 , βˆ’ 3 ⟩
  • C ⟨ 4 , 1 . 9 ⟩
  • D ⟨ βˆ’ 3 , 1 . 9 ⟩
  • E ⟨ βˆ’ 3 , βˆ’ 1 . 7 ⟩

Q24:

Given that the coordinates of and are and respectively, find .

Q25:

Consider the vector in the given diagram.

What are the coordinates of its terminal point?

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

What are the coordinates of its initial point?

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

What are the components of the vector?

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

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