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 ||=10A and ||=17B, where the measure of the angle between A and B is 120, find (+)(2)ABAB correct to the nearest hundredth.

Q2:

Suppose 𝐴𝐵𝐶𝐷 is a square of side 47. Determine 𝐴𝐵𝐴𝐶.

Q3:

Suppose a rectangle 𝐴𝐵𝐶𝐷 with 𝐴𝐵=32 and 𝐵𝐶=11. Determine 𝐴𝐶𝐵𝐷.

Q4:

Find the value of 𝜃 approximated to the nearest hundredth.

Q5:

AB=.

  • A||||AB
  • B||+||AB
  • C||||𝜃ABtan
  • D||||𝜃ABcos
  • E||||𝜃ABsin

Q6:

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

  • Aperpendicular
  • Bparallel

Q7:

Consider a square 𝐴𝐵𝐶𝐷 of side 32.6. Determine 𝐴𝐷𝐴𝐶.

Q8:

In a rectangle 𝐴𝐵𝐶𝐷, 𝐴𝐵=7.7 and 𝐵𝐶=3.8. Determine 5𝐴𝐵𝐴𝐶.

Q9:

If 𝐴𝐵𝐶𝐷𝐸𝐹 is a regular hexagon whose side is 7.1, find 𝐶𝐴+𝐴𝐹𝐴𝐷.

Q10:

If 𝐴𝐵𝐶𝐷𝐸𝐹 is a regular hexagon whose side is 9.6, find 𝐴𝐵𝐸𝐹𝐴𝐷.

Q11:

Triangle 𝐴𝐵𝐶 has a right angle at 𝐵. Given 𝐴𝐵=20, 𝐵𝐶=5, and that 𝐷 is the midpoint of 𝐴𝐶. Determine 𝐷𝐵75𝐵𝐶 to the nearest hundredth.

Q12:

Triangle 𝐴𝐵𝐶 has a right angle at 𝐵. Given 𝐴𝐵=38.6, 𝐵𝐶=36.4, and that 𝐷 is the midpoint of 𝐴𝐶, determine 𝐵𝐴𝐵𝐷 to the nearest hundredth.

Q13:

Triangle 𝐴𝐵𝐶 has a right angle at 𝐵. Given 𝐴𝐵=31.2 and 𝐵𝐶=20.4, determine 𝐴𝐵𝐴𝐶.

Q14:

If Aij=15+5 and Bij=8+𝑘, find all the possible values of 𝑘 that make the measure of the angle between the two vectors 135.

  • A𝑘=4 or 𝑘=16
  • B𝑘=16 or 𝑘=11
  • C𝑘=3 or 𝑘=20
  • D𝑘=11 or 𝑘=3

Q15:

Let A=28 and B=32. Find the angle between A and B giving your answer to two decimal places.

Q16:

Given that Aij=9+3, and Bij=38, determine the angle between the two vectors rounded to the nearest minute.

  • A9252
  • B8753
  • C27
  • D13447

Q17:

If ABAB=3|×|, and 𝜃 is the angle between A and B, find the value of 𝜃 in the range 0<𝜃<180.

Q18:

If ||=16A, ||=10B, and the measure of the angle between A and B is 30, calculate AB.

  • A133
  • B803
  • C133
  • D80
  • E803

Q19:

If ABAB=|×|, find the value of 𝜃 in the range 0<𝜃<180,where 𝜃 is the angle between A and B.

Q20:

If 𝜃 is the angle between A and B and ABAB=13|×|, find the value of 𝜃 in the range 0<𝜃<180.

Q21:

Given that ||=||AB, AB=69, and the measure of the angle between A and B is 60, determine ||A to the nearest hundredth.

Q22:

Between unit vectors A and B is an angle of 66. Find AB to the nearest tenth.

Q23:

If 𝜃 is the smaller angle between the two vectors A and B, find cos𝜃.

  • AABB||
  • BABAB+||||
  • CABAB||+||
  • DABA||
  • EABAB||||

Q24:

Given that ||=12A, ||=25B, and ||=23AB, determine the measure of the angle between A and B rounded to the nearest minute.

  • A6625
  • B8649
  • C8117
  • D2219

Q25:

Given that ||=9A, ||=9B, and ||=5AB, determine |+|AB, and give the measure of the angle between A and B to the nearest hundredth.

  • A|+|=15.18AB, 𝜃=32.26
  • B|+|=17.29AB, 𝜃=85.40
  • C|+|=12.73AB, 𝜃=57.74
  • D|+|=17.29AB, 𝜃=32.26

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