Worksheet: The Angle between Two Vectors in the Coordinate Plane

In this worksheet, we will practice using the dot product to find the angle between two nonzero vectors in a plane.

Q1:

If | | = 1 0 A and | | = 1 7 B , where the measure of the angle between A and B is 1 2 0 , find ( + ) ( 2 ) A B A B correct to the nearest hundredth.

Q2:

Suppose 𝐴 𝐵 𝐶 𝐷 is a square of side 47. Determine 𝐴 𝐵 𝐴 𝐶 .

Q3:

Suppose a rectangle 𝐴 𝐵 𝐶 𝐷 with 𝐴 𝐵 = 3 2 and 𝐵 𝐶 = 1 1 . Determine 𝐴 𝐶 𝐵 𝐷 .

Q4:

Find the value of 𝜃 approximated to the nearest hundredth.

Q5:

A B = .

  • A | | | | A B
  • B | | + | | A B
  • C | | | | 𝜃 A B c o s
  • D | | | | 𝜃 A B t a n
  • E | | | | 𝜃 A B s i n

Q6:

If A and B satisfy A B = 0 , how are the two vectors related?

  • Aperpendicular
  • Bparallel

Q7:

Suppose square 𝐴 𝐵 𝐶 𝐷 of side 32.6. Determine 𝐴 𝐷 𝐴 𝐶 .

Q8:

In a rectangle 𝐴 𝐵 𝐶 𝐷 , 𝐴 𝐵 = 7 . 7 and 𝐵 𝐶 = 3 . 8 . Determine 5 𝐴 𝐵 𝐴 𝐶 .

Q9:

If 𝐴 𝐵 𝐶 𝐷 𝐸 𝐹 is a regular hexagon whose side is 7.1, find 𝐶 𝐴 + 𝐴 𝐹 𝐴 𝐷 .

Q10:

If 𝐴 𝐵 𝐶 𝐷 𝐸 𝐹 is a regular hexagon whose side is 9.6, find 𝐴 𝐵 𝐸 𝐹 𝐴 𝐷 .

Q11:

Triangle 𝐴 𝐵 𝐶 has a right angle at 𝐵 . Given 𝐴 𝐵 = 2 0 , 𝐵 𝐶 = 5 , and that 𝐷 is the midpoint of 𝐴 𝐶 . Determine 𝐷 𝐵 7 5 𝐵 𝐶 to the nearest hundredth.

Q12:

Triangle 𝐴 𝐵 𝐶 has a right angle at 𝐵 . Given 𝐴 𝐵 = 3 8 . 6 , 𝐵 𝐶 = 3 6 . 4 , and that 𝐷 is the midpoint of 𝐴 𝐶 , determine 𝐵 𝐴 𝐵 𝐷 to the nearest hundredth.

Q13:

Triangle 𝐴 𝐵 𝐶 has a right angle at 𝐵 . Given 𝐴 𝐵 = 3 1 . 2 and 𝐵 𝐶 = 2 0 . 4 , determine 𝐴 𝐵 𝐴 𝐶 .

Q14:

If A i j = 1 5 + 5 and B i j = 8 + 𝑘 , find all the possible values of 𝑘 that make the measure of the angle between the two vectors 1 3 5 .

  • A 𝑘 = 4 or 𝑘 = 1 6
  • B 𝑘 = 3 or 𝑘 = 2 0
  • C 𝑘 = 1 6 or 𝑘 = 1 1
  • D 𝑘 = 1 1 or 𝑘 = 3

Q15:

Let A = 2 8 and B = 3 2 . Find the angle between A and B giving your answer to two decimal places.

Q16:

Given that A i j = 9 + 3 , and B i j = 3 8 , determine the angle between the two vectors rounded to the nearest minute.

  • A 8 7 5 3
  • B 9 2 5 2
  • C 1 3 4 4 7
  • D 2 7

Q17:

If A B A B = 3 | × | , and 𝜃 is the angle between A and B , find the value of 𝜃 in the range 0 < 𝜃 < 1 8 0 .

Q18:

If | | = 1 6 A , | | = 1 0 B , and the measure of the angle between A and B is 3 0 , calculate A B .

  • A 8 0 3
  • B 1 3 3
  • C 1 3 3
  • D80
  • E 8 0 3

Q19:

If A B A B = | × | , find the value of 𝜃 in the range 0 < 𝜃 < 1 8 0 ,where 𝜃 is the angle between A and B .

Q20:

If 𝜃 is the angle between A and B and A B A B = 1 3 | × | , find the value of 𝜃 in the range 0 < 𝜃 < 1 8 0 .

Q21:

Given that | | = | | A B , A B = 6 9 , and the measure of the angle between A and B is 6 0 , determine | | A to the nearest hundredth.

Q22:

Between unit vectors A and B is an angle of 6 6 . Find A B to the nearest tenth.

Q23:

If 𝜃 is the smaller angle between the two vectors A and B , find c o s 𝜃 .

  • A A B A B | | + | |
  • B A B B | |
  • C A B A | |
  • D A B A B | | | |
  • E A B A B + | | | |

Q24:

Given that | | = 1 2 A , | | = 2 5 B , and | | = 2 3 A B , determine the measure of the angle between A and B rounded to the nearest minute.

  • A 6 6 2 5
  • B 2 2 1 9
  • C 8 1 1 7
  • D 8 6 4 9

Q25:

Given that | | = 9 A , | | = 9 B , and | | = 5 A B , determine | + | A B , and give the measure of the angle between A and B to the nearest hundredth.

  • A | + | = 1 7 . 2 9 A B , 𝜃 = 3 2 . 2 6
  • B | + | = 1 2 . 7 3 A B , 𝜃 = 5 7 . 7 4
  • C | + | = 1 7 . 2 9 A B , 𝜃 = 8 5 . 4 0
  • D | + | = 1 5 . 1 8 A B , 𝜃 = 3 2 . 2 6

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