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 ||=9.5A, ||=15.0B, and ABC×=122, where C is a unit vector, find the measure of the angle 𝜃 between A and B rounded to the nearest minute given 0<𝜃<90∘∘.

  • A 3 7 3 3 ′ ∘
  • B 5 8 5 3 ′ ∘
  • C 5 2 4 3 ′ ∘
  • D 3 1 7 ′ ∘
  • E 1 2 1 7 ′ ∘

Q2:

Given 𝐴(−3,−5,6), 𝐵(0,3,−7), 𝐶(−8,10,−2), and 𝐷(−3,9,−6), determine the measure of the angle between vectors 𝐴𝐵 and 𝐶𝐷 rounded to the nearest hundredth.

Q3:

Find |×|⋅ABAB.

  • A t a n 𝜃
  • B s i n 𝜃
  • C0
  • D1
  • E c o s 𝜃

Q4:

If Aik=2+5 and Bijk=4+3+, find the measure of the angle between the two vectors rounded to the nearest hundredth.

Q5:

Given that Aijk=4−−2 and B=⟨2,−2,4⟩, determine, to the nearest hundredth, the measure of the smaller angle between the two vectors.

Q6:

Given that ||=35A, ||=23B, and AB⋅=−805√22, determine the measure of the smaller angle between the two vectors.

Q7:

The angle between A and B is 22∘. If ||=3||=25.2AB, find AB⋅ to the nearest hundredth.

Q8:

Find the angle 𝜃 between the vectors V=⟨5,1,−2⟩ and W=⟨4,−4,3⟩. Give your answer correct to two decimal places.

Q9:

Find the angle 𝜃 between the vectors Vi= and Wijk=3+2+4. Give your answer correct to two decimal places.

Q10:

Find the angle 𝜃 between the vectors V=⟨2,1,4⟩ and W=⟨1,−2,0⟩.

Q11:

Find the angle 𝜃 between the vectors V=⟨4,2,−1⟩ and W=⟨8,4,−2⟩.

Q12:

Find the angle 𝜃 between the vectors Vijk=−+2+ and Wijk=−3+6+3.

Q13:

Find the angle 𝜃 between the vectors V=⟨7,2,−10⟩ and W=⟨2,6,4⟩. Give your answer correct to one decimal place.

Q14:

Find the value of 𝑥 given A=4𝜃,𝑥,𝜃coslogsin, B=𝜃,16,4𝜃coslogsin and AB⋅=10 where 𝜃 is the angle between the vectors A and B. Give the answer to two decimal places.

Q15:

Given that Aijk=−4+7+6 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 𝐴𝐵 and 𝐶𝐷 rounded to the nearest hundredth.

Q17:

Given 𝐴(6,9,10), 𝐵(0,−5,6), 𝐶(−10,−10,4), and 𝐷(−7,−6,2), determine the measure of the angle between vectors 𝐴𝐵 and 𝐶𝐷 rounded to the nearest hundredth.

Q18:

If Aik=6+ and Bijk=2+4+7, find the measure of the angle between the two vectors rounded to the nearest hundredth.

Q19:

Determine the angle between the vectors 21−1 and 10−1 in ℝ.

  • A a r c c o s √ 3 2
  • B √ 3 2
  • C0
  • D 𝜋 2

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