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 𝐴 = 1 0 and 𝐵 = 1 7 , where the measure of the angle between 𝐴 and 𝐵 is 1 2 0 , find 𝐴 + 𝐵 𝐴 2 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 𝜃 s i n
  • C 𝐴 + 𝐵
  • D 𝐴 𝐵 𝜃 c o s
  • E 𝐴 𝐵 𝜃 t a 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 = 1 5 𝑖 + 5 𝑗 and 𝐵 = 8 𝑖 + 𝑘 𝑗 , find all the possible values of 𝑘 that make the measure of the angle between the two vectors 1 3 5 .

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

Q15:

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

Q16:

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

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

Q17:

If 𝐴 𝐵 = 3 𝐴 × 𝐵 , find the value of 𝜃 in the range 0 < 𝜃 < 1 8 0 .

Q18:

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

  • A80
  • B 8 0 3
  • C 1 3 3
  • D 8 0 3
  • E 1 3 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 = 1 3 𝐴 × 𝐵 , 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 𝐵
  • B 𝐴 𝐵 𝐴
  • C 𝐴 𝐵 𝐴 + 𝐵
  • D 𝐴 𝐵 𝐴 𝐵
  • E 𝐴 + 𝐵 𝐴 𝐵

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