Vectors - Problems - Angle between unit vectors

• In this lecture, we will discuss problems related to vectors and specifically focus on finding the angle between unit vectors.
• Unit vectors are vectors with magnitude equal to 1.
• The formula to find the angle between two unit vectors u and v is given by:
  • cosθ = u · v, where · represents the dot product.
• Let’s solve some examples to understand better.

Example 1

• Given u = (1, 0, 0) and v = (0, 1, 0), find the angle between the two unit vectors. Solution:
• To find the angle between u and v, we can use the formula: cosθ = u · v = (1 * 0) + (0 * 1) + (0 * 0) = 0 + 0 + 0 = 0
• Therefore, the angle between u and v is θ = cos^(-1)(0) = 90°.

Example 2

• Consider u = (2, -1, 3) and v = (1, 4, -2). Find the angle between the two unit vectors. Solution:
• To find the angle between u and v, we can use the formula: cosθ = u · v = (2 * 1) + (-1 * 4) + (3 * -2) = 2 - 4 - 6 = -8
• Therefore, the angle between u and v is θ = cos^(-1)(-8) = 140.53°.

Example 3

• Let u = (3, 4, 5) and v = (-1, -2, -3). Determine the angle between the two unit vectors. Solution:
• To find the angle between u and v, we can use the formula: cosθ = u · v = (3 * -1) + (4 * -2) + (5 * -3) = -3 - 8 - 15 = -26
• Therefore, the angle between u and v is θ = cos^(-1)(-26) = 155.28°.

Example 4

• Given u = (sqrt(2)/2, sqrt(2)/2) and v = (-sqrt(2)/2, sqrt(2)/2), determine the angle between the two unit vectors. Solution:
• To find the angle between u and v, we can use the formula: cosθ = u · v = (sqrt(2)/2 * -sqrt(2)/2) + (sqrt(2)/2 * sqrt(2)/2) = -1/2 + 1/2 = 0
• Therefore, the angle between u and v is θ = cos^(-1)(0) = 90°.

Example 5

• Consider u = (7, -3) and v = (4, 1). Find the angle between the two unit vectors. Solution:
• To find the angle between u and v, we can use the formula: cosθ = u · v = (7 * 4) + (-3 * 1) = 28 - 3 = 25
• Therefore, the angle between u and v is θ = cos^(-1)(25) which is not defined since the cosine function is only defined between -1 and 1.

Vectors - Problems - Angle between unit vectors

Slide s 11-20

Slide 11

• Example 6:
  • Given u = (3, -4) and v = (-6, 8), find the angle between the two unit vectors. Solution:
  • To find the angle between u and v, we can use the formula: cosθ = u · v = (3 * -6) + (-4 * 8) = -18 - 32 = -50
  • Therefore, the angle between u and v is θ = cos^(-1)(-50) which is not defined since the cosine function is only defined between -1 and 1.

Slide 12

• Example 7:
  • Consider u = (1, 2, -1) and v = (-2, 4, -2). Determine the angle between the two unit vectors. Solution:
  • To find the angle between u and v, we can use the formula: cosθ = u · v = (1 * -2) + (2 * 4) + (-1 * -2) = -2 + 8 + 2 = 8
  • Therefore, the angle between u and v is θ = cos^(-1)(8) which is not defined since the cosine function is only defined between -1 and 1.

Slide 13

• Example 8:
  • Given u = (1, 1, -4) and v = (2, -3, 5), find the angle between the two unit vectors. Solution:
  • To find the angle between u and v, we can use the formula: cosθ = u · v = (1 * 2) + (1 * -3) + (-4 * 5) = 2 - 3 - 20 = -21
  • Therefore, the angle between u and v is θ = cos^(-1)(-21) which is not defined since the cosine function is only defined between -1 and 1.

Slide 14

• Example 9:
  • Consider u = (2, -1, 1) and v = (-3, 1, -2). Find the angle between the two unit vectors. Solution:
  • To find the angle between u and v, we can use the formula: cosθ = u · v = (2 * -3) + (-1 * 1) + (1 * -2) = -6 - 1 - 2 = -9
  • Therefore, the angle between u and v is θ = cos^(-1)(-9) which is not defined since the cosine function is only defined between -1 and 1.

Slide 15

• Example 10:
  • Given u = (4, -2, 0) and v = (0, 1, 3), determine the angle between the two unit vectors. Solution:
  • To find the angle between u and v, we can use the formula: cosθ = u · v = (4 * 0) + (-2 * 1) + (0 * 3) = 0 - 2 + 0 = -2
  • Therefore, the angle between u and v is θ = cos^(-1)(-2) which is not defined since the cosine function is only defined between -1 and 1.

Slide 16

• Recap:
  • In this lecture, we discussed problems related to vectors and specifically focused on finding the angle between unit vectors.
  • The formula to find the angle between two unit vectors u and v is given by:
    • cosθ = u · v, where · represents the dot product.
  • We solved several examples to understand the concept better.

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