title: Vectors - Problems - Condition for Obtuse Angle Between Vectors

Vectors - Problems - Condition for Obtuse Angle Between Vectors

Problem

Find the condition for an obtuse angle between two vectors a and b.

  • a = (a₁, a₂, a₃)
  • b = (b₁, b₂, b₃)

Solution

  • The angle between two vectors a and b can be found using the dot product formula:
    • a · b = |a| |b| cos θ, where θ is the angle between a and b
  • To find the condition for an obtuse angle, we need to consider the sign of cos θ.
  • Cosine of an obtuse angle is negative.
  • Therefore, the condition for an obtuse angle between two vectors is:
    • a · b < 0

Example

Find the condition for an obtuse angle between the vectors a = (1, -2, 3) and b = (4, 5, -6).

Solution

  • Dot product of a and b = a · b = (1)(4) + (-2)(5) + (3)(-6) = 4 - 10 - 18 = -24
  • The condition for an obtuse angle between a and b is a · b < 0.

Equation

The condition for an obtuse angle between two vectors a and b can be written as: a · b < 0

Summary

  • The condition for an obtuse angle between two vectors a and b is a · b < 0.
  • This condition can be obtained by using the dot product formula.
  • An obtuse angle has a negative cosine value.
  • Hence, the dot product of the vectors must be negative for an obtuse angle.

End of Slides

title: Vectors - Problems - Determining Collinearity of Vectors

Vectors - Problems - Determining Collinearity of Vectors

Problem

Given three vectors a, b, and c, determine whether they are collinear.

  • a = (a₁, a₂, a₃)
  • b = (b₁, b₂, b₃)
  • c = (c₁, c₂, c₃)

Solution

  • Three vectors are collinear if they are scalar multiples of each other.
  • To check if vectors a, b, and c are collinear, we need to verify the following condition:
    • a = k b = k c
  • Here, k is a scalar value.

Example

Given three vectors a = (2, -1, 3), b = (4, -2, 6), and c = (-6, 3, -9), determine whether they are collinear.

Solution

  • The vectors a, b, and c are collinear if they satisfy the condition: a = k b = k c
  • Let’s find the value of k by comparing the corresponding components of the vectors.
  • a₁ = 2, b₁ = 4, c₁ = -6 => 2 = k(4) = k(-6), k = 1/2 = -1/3
  • a₂ = -1, b₂ = -2, c₂ = 3 => -1 = k(-2) = k(3), k = 1/2 = -1/3
  • a₃ = 3, b₃ = 6, c₃ = -9 => 3 = k(6) = k(-9), k = 1/2 = -1/3
  • The values of k obtained for all three components are the same. Hence, the vectors a, b, and c are collinear.

Equation

The condition for collinearity of three vectors a, b, and c can be written as: a = k b = k c

Summary

  • Three vectors are collinear if they are scalar multiples of each other.
  • To check for collinearity, compare the components of the vectors and find a common scalar value.
  • If the vectors satisfy the condition a = k b = k c, they are collinear.
  • Collinear vectors lie on the same line or are parallel to each other.

End of Slides

title: Vectors - Problems - Condition for Obtuse Angle Between Vectors

Problem

Find the condition for an obtuse angle between two vectors a and b.

  • a = (a₁, a₂, a₃)
  • b = (b₁, b₂, b₃)

Solution

  • The angle between two vectors a and b can be found using the dot product formula:
    • a · b = |a| |b| cos θ, where θ is the angle between a and b
  • To find the condition for an obtuse angle, we need to consider the sign of cos θ.
  • Cosine of an obtuse angle is negative.
  • Therefore, the condition for an obtuse angle between two vectors is:
    • a · b < 0

Example

Find the condition for an obtuse angle between the vectors a = (1, -2, 3) and b = (4, 5, -6).

Solution

  • Dot product of a and b = a · b = (1)(4) + (-2)(5) + (3)(-6) = 4 - 10 - 18 = -24
  • The condition for an obtuse angle between a and b is a · b < 0.

Equation

The condition for an obtuse angle between two vectors a and b can be written as: a · b < 0

Summary

  • The condition for an obtuse angle between two vectors a and b is a · b < 0.
  • This condition can be obtained by using the dot product formula.
  • An obtuse angle has a negative cosine value.
  • Hence, the dot product of the vectors must be negative for an obtuse angle.

End of Slides

title: Vectors - Problems - Determining Collinearity of Vectors

Problem

Given three vectors a, b, and c, determine whether they are collinear.

  • a = (a₁, a₂, a₃)
  • b = (b₁, b₂, b₃)
  • c = (c₁, c₂, c₃)

Solution

  • Three vectors are collinear if they are scalar multiples of each other.
  • To check if vectors a, b, and c are collinear, we need to verify the following condition:
    • a = k b = k c
  • Here, k is a scalar value.

Example

Given three vectors a = (2, -1, 3), b = (4, -2, 6), and c = (-6, 3, -9), determine whether they are collinear.

Solution

  • The vectors a, b, and c are collinear if they satisfy the condition: a = k b = k c
  • Let’s find the value of k by comparing the corresponding components of the vectors.
  • a₁ = 2, b₁ = 4, c₁ = -6 => 2 = k(4) = k(-6), k = 1/2 = -1/3
  • a₂ = -1, b₂ = -2, c₂ = 3 => -1 = k(-2) = k(3), k = 1/2 = -1/3
  • a₃ = 3, b₃ = 6, c₃ = -9 => 3 = k(6) = k(-9), k = 1/2 = -1/3
  • The values of k obtained for all three components are the same. Hence, the vectors a, b, and c are collinear.

Equation

The condition for collinearity of three vectors a, b, and c can be written as: a = k b = k c

Summary

  • Three vectors are collinear if they are scalar multiples of each other.
  • To check for collinearity, compare the components of the vectors and find a common scalar value.
  • If the vectors satisfy the condition a = k b = k c, they are collinear.
  • Collinear vectors lie on the same line or are parallel to each other.

End of Slides