Worksheet: Polar Form of a Vector

In this worksheet, we will practice converting between rectangular and polar forms of a vector.

Q1:

If 𝑂𝐴=7,60 is the position vector, in polar form, of the point 𝐴 relative to the origin 𝑂, find the 𝑥𝑦-coordinates of 𝐴.

  • A 7 3 2 , 7 2
  • B 7 2 , 7 3 2
  • C 7 3 2 , 7 3 2
  • D 7 2 , 7 2

Q2:

If 𝑂𝐶=43,3𝜋4 is the position vector, in polar form, of the point 𝐶 relative to the origin 𝑂, find the 𝑥𝑦-coordinates of the point 𝐶.

  • A 2 6 , 4 6
  • B 2 6 , 2 6
  • C 2 6 , 2 6
  • D 2 2 , 2 2
  • E 4 6 , 2 6

Q3:

Given the point 𝐴43,4, express, in polar form, its position vector relative to the origin point.

  • A 8 , 1 1 𝜋 3
  • B 8 , 1 1 𝜋 6
  • C 8 2 , 1 1 𝜋 1 2
  • D 8 , 1 1 𝜋 1 2

Q4:

Given that A=6,15, B=𝑘,10, and AB, find the value of 𝑘.

Q5:

Trapezium 𝐴𝐵𝐶𝐷 has vertices 𝐴(10,11), 𝐵(𝑘,8), 𝐶(4,12), and 𝐷(2,6). Given that 𝐴𝐵𝐶𝐷, find the value of 𝑘.

Q6:

Let A=32 and B=57.5.

Find AB.

Which of the following is, therefore, true of the vectors?

  • AThey are parallel and in the same direction.
  • BIt does not tell anything about the vectors.
  • CThey are perpendicular.
  • DThe two vectors are equal in length.
  • EThey are parallel but in opposite directions.

Q7:

Given that the vectors A=62 and B=3𝑥 are perpendicular, find the value of 𝑥.

Q8:

Given that the vectors A=3𝑥+1 and B=2𝑥3 are perpendicular, find the value of 𝑥.

Q9:

Given the point 𝐴33,9, express, in polar form, its position vector relative to the origin point.

  • A 6 3 , 1 0 𝜋 3
  • B 1 2 , 1 0 𝜋 3
  • C 6 3 , 5 𝜋 3
  • D 6 , 5 𝜋 6
  • E 1 2 , 5 𝜋 6

Q10:

Given the point 𝐴23,6, express, in polar form, its position vector relative to the origin point.

  • A 4 3 , 4 𝜋 3
  • B 8 , 4 𝜋 3
  • C 4 3 , 2 𝜋 3
  • D 4 , 𝜋 3
  • E 8 , 𝜋 3

Q11:

Given the point 𝐴33,9, express, in polar form, its position vector relative to the origin point.

  • A 6 3 , 8 𝜋 3
  • B 6 , 8 𝜋 3
  • C 6 3 , 4 𝜋 3
  • D 1 2 , 2 𝜋 3
  • E 6 , 2 𝜋 3

Q12:

Given the point 𝐴23,2, express, in polar form, its position vector relative to the origin point.

  • A 4 , 7 𝜋 3
  • B 4 2 , 7 𝜋 3
  • C 4 , 7 𝜋 6
  • D 4 , 7 𝜋 1 2
  • E 4 2 , 7 𝜋 1 2

Q13:

Consider the vector 75ij. Calculate the direction of the vector, giving your solution as an angle to the nearest degree measured counterclockwise from the positive 𝑥-axis.

  • A 2 1 5
  • B 3 5
  • C 3 2 4
  • D 2 1 6
  • E 3 6

Q14:

Consider the vector 23. Calculate the direction of the vector, giving your solution as an angle to the nearest degree measured counterclockwise from the positive 𝑥-axis.

Q15:

Consider the vector 𝑣 with modulus 3 at an angle of 45 above the positive 𝑥-axis. Using trigonometry, calculate the 𝑥- and 𝑦-components of the vector and, hence, write 𝑣 in the form 𝑥𝑦. Round your answer to three significant figures.

  • A 2 . 1 0 2 . 1 0
  • B 2 . 1 2 2 . 1 2
  • C 2 . 2 0 2 . 2 0
  • D 2 . 1 1 2 . 1 1
  • E 2 . 1 3 2 . 1 3

Q16:

Consider the vector v=32.

Which of the following graphs accurately represents the vector?

  • A
  • B
  • C
  • D
  • E

Calculate the modulus of the vector.

  • A 2 6
  • B 1 3
  • C26
  • D1
  • E13

Given that positive numbers represent measuring counter-clockwise, calculate the measure of the angle the vector makes with the positive 𝑥-axis. Give your answer to 3 significant figures between 180 and 180.

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