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.

  • A732,72
  • B72,732
  • C732,732
  • D72,72

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.

  • A26,46
  • B26,26
  • C26,26
  • D22,22
  • E46,26

Q3:

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

  • A8,11𝜋3
  • B8,11𝜋6
  • C82,11𝜋12
  • D8,11𝜋12

Q4:

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

Q5:

Trapezoid 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.

  • A63,10𝜋3
  • B12,10𝜋3
  • C63,5𝜋3
  • D6,5𝜋6
  • E12,5𝜋6

Q10:

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

  • A43,4𝜋3
  • B8,4𝜋3
  • C43,2𝜋3
  • D4,𝜋3
  • E8,𝜋3

Q11:

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

  • A63,8𝜋3
  • B6,8𝜋3
  • C63,4𝜋3
  • D12,2𝜋3
  • E6,2𝜋3

Q12:

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

  • A4,7𝜋3
  • B42,7𝜋3
  • C4,7𝜋6
  • D4,7𝜋12
  • E42,7𝜋12

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.

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.

  • A2.102.10
  • B2.122.12
  • C2.202.20
  • D2.112.11
  • E2.132.13

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.

  • A26
  • B13
  • C26
  • D1
  • E13

Given that positive numbers represent measuring counterclockwise, 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.

Q17:

If the force 𝐹=6newtons acts in the direction 60 east of north, then F=.

  • A333ij
  • B33+3ij
  • C333ij
  • D3+33ij

Q18:

If A=7,5𝜋3, then vector A, in terms of the fundamental unit vectors, equals .

  • A72+732ij
  • B72732ij
  • C732+72ij
  • D732+72ij

Q19:

If Aij=, then the polar form of A is .

  • A2,5𝜋4
  • B2,𝜋4
  • C2,7𝜋4
  • D2,3𝜋4

Q20:

If 𝑃𝑀=5,53, then the polar form of vector 𝑃𝑀 is .

  • A10,𝜋6
  • B10,𝜋2
  • C(10,𝜋)
  • D10,𝜋3

Q21:

If A=(1,3) and rotates around the origin with an angle of measure 30 anticlockwise, then the polar form of vector A after rotation is .

  • A(2,60)
  • B(4,90)
  • C(2,90)
  • D(2,30)

Q22:

If ||=6Acm, then A=.

  • A6,63
  • B3,3
  • C33,3
  • D3,33

Q23:

If the polar form of 𝑂𝐴 is 10,𝜋3, then the polar form of 𝐴𝑂 is .

  • A10,𝜋3
  • B5,𝜋3
  • C10,2𝜋3
  • D10,4𝜋3

Q24:

If Aij=3+4, Bj=4, and C=6,𝜋10, then ||+||+||=ABC.

  • A6
  • B11
  • C10
  • D15

Q25:

If A=(8,𝜋) and Bij=4+5, then 𝐴𝐵=.

  • A(4,5)
  • B(4,5)
  • C(4,5)
  • D(4,5)

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