Lesson: The Angle between Two Vectors in the Coordinate Plane

In this lesson, we will learn how to use the dot product to find the angle between two nonzero vectors in a plane.

Sample Question Videos

  • 04:46
  • 06:45

Worksheet: The Angle between Two Vectors in the Coordinate Plane • 23 Questions • 2 Videos

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 = .

Q6:

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

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 ∘ .

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.

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 .

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 .
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