Worksheet: Angle between Two Vectors in Space

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In this worksheet, we will practice finding the angle between two vectors in space using their dot product.

Q1:

If , , and , where is a unit vector, find the measure of the angle between and rounded to the nearest minute given .

  • A
  • B
  • C
  • D
  • E

Q2:

Given 𝐴 ( − 3 , − 5 , 6 ) , 𝐵 ( 0 , 3 , − 7 ) , 𝐶 ( − 8 , 1 0 , − 2 ) , and 𝐷 ( − 3 , 9 , − 6 ) , determine the measure of the angle between vectors 𝐴 𝐵 and 𝐶 𝐷 rounded to the nearest hundredth.

Q3:

Find ‖ ‖ ⃑ 𝐴 × ⃑ 𝐵 ‖ ‖ ⃑ 𝐴 ⋅ ⃑ 𝐵 .

  • A0
  • B s i n 𝜃
  • C1
  • D t a n 𝜃
  • E c o s 𝜃

Q4:

If ⃑ 𝐴 = 2 ⃑ 𝑖 + 5 ⃑ 𝑘 and ⃑ 𝐵 = 4 ⃑ 𝑖 + 3 ⃑ 𝑗 + ⃑ 𝑘 , find the measure of the angle between the two vectors rounded to the nearest hundredth.

Q5:

Given that ⃑ 𝐴 = 4 ⃑ 𝑖 − ⃑ 𝑗 − 2 ⃑ 𝑘 and ⃑ 𝐵 = ( 2 , − 2 , 4 ) , determine, to the nearest hundredth, the measure of the smaller angle between the two vectors.

Q6:

Given that ‖ ‖ ⃑ 𝐴 ‖ ‖ = 3 5 , ‖ ‖ ⃑ 𝐵 ‖ ‖ = 2 3 , and ⃑ 𝐴 ⋅ ⃑ 𝐵 = − 8 0 5 √ 2 2 , determine the measure of the smaller angle between the two vectors.

Q7:

The angle between ⃑ 𝐴 and ⃑ 𝐵 is 2 2 ∘ . If ‖ ‖ ⃑ 𝐴 ‖ ‖ = 3 ‖ ‖ ⃑ 𝐵 ‖ ‖ = 2 5 . 2 , find ⃑ 𝐴 ⋅ ⃑ 𝐵 to the nearest hundredth.

Q8:

Find the angle between the vectors and . Give your answer correct to two decimal places.

Q9:

Find the angle between the vectors and . Give your answer correct to two decimal places.

Q10:

Find the angle between the vectors and .

Q11:

Find the angle between the vectors and .

Q12:

Find the angle between the vectors and .

Q13:

Find the angle between the vectors and . Give your answer correct to one decimal place.

Q14:

Find the value of 𝑥 given ⃑ 𝐴 = 4 𝜃 , 𝑥 , 𝜃 c o s l o g s i n 3 , ⃑ 𝐵 = 𝜃 , 1 6 , 4 𝜃 c o s l o g s i n 2 and ⃑ 𝐴 ⋅ ⃑ 𝐵 = 1 0 where 𝜃 is the angle between the vectors ⃑ 𝐴 and ⃑ 𝐵 . Give the answer to two decimal places.

Q15:

Given that ⃑ 𝐴 = − 4 ⃑ 𝑖 + 7 ⃑ 𝑗 + 6 ⃑ 𝑘 and ⃑ 𝐵 = ( 1 , 4 , 2 ) , determine, to the nearest hundredth, the measure of the smaller angle between the two vectors.

Q16:

Given 𝐴 ( − 7 , 4 , 0 ) , 𝐵 ( − 7 , 1 , − 9 ) , 𝐶 ( 4 , 5 , 8 ) , and 𝐷 ( 1 , − 6 , 4 ) , determine the measure of the angle between vectors 𝐴 𝐵 and 𝐶 𝐷 rounded to the nearest hundredth.

Q17:

Given 𝐴 ( 6 , 9 , 1 0 ) , 𝐵 ( 0 , − 5 , 6 ) , 𝐶 ( − 1 0 , − 1 0 , 4 ) , and 𝐷 ( − 7 , − 6 , 2 ) , determine the measure of the angle between vectors 𝐴 𝐵 and 𝐶 𝐷 rounded to the nearest hundredth.

Q18:

If ⃑ 𝐴 = 7 ⃑ 𝑖 + 2 ⃑ 𝑗 + 4 ⃑ 𝑘 and ⃑ 𝐵 = 5 ⃑ 𝑖 + ⃑ 𝑘 , find the measure of the angle between the two vectors rounded to the nearest hundredth.

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