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

Q1:

.

Q2:

If and satisfy , how are the two vectors related?

Q3:

If and are two perpendicular vectors, then .

Q4:

If and , where the size of the angle between and is , find to the nearest hundredth.

Q5:

Q6:

Q7:

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

Q8:

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

Q9:

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

Q10:

In rectangle 𝐴 𝐵 𝐶 𝐷 , we have 𝐴 𝐵 = 1 5 and 𝐵 𝐶 = 1 1 . Determine 𝐵 𝐶 ⋅ 5 𝐷 𝐵 to the nearest hundredth.

Q11:

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

Q12:

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

Q13:

If 𝐴 𝐵 𝐶 𝐷 𝐸 𝐹 is a regular hexagon whose side is 17.2, find 𝐶 𝐴 + 𝐴 𝐹 ⋅ 𝐴 𝐷 .

Q14:

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

Q15:

If 𝐴 𝐵 𝐶 𝐷 𝐸 𝐹 is a regular hexagon whose side is 32.3, find 𝐴 𝐵 − 𝐸 𝐹 ⋅ 𝐴 𝐷 .

Q16:

If 𝐴 𝐵 𝐶 𝐷 𝐸 𝐹 is a regular hexagon whose side is 32.6, find 𝐴 𝐵 − 𝐸 𝐹 ⋅ 𝐴 𝐷 .

Q17:

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.

Q18:

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.

Q19:

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 .

Q20:

If and , find all the possible values of that make the size of the angle between the two vectors .

Q21:

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

Q22:

Given that , and , determine the angle between the two vectors rounded to the nearest minute.

Q23:

If , find the value of in the range .

Q24:

If , , and the size of the angle between and is , calculate .

Q25:

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