Lecture on Dot Product of Vectors

What is Dot Product?

• Multiplication of two vectors, denoted by a dot between them.
• Example: Dot product of vectors a and b.
• Result is a scalar (a number).

Calculating Dot Product

• If p = (a, b, c) and q = (x, y, z), then:
  • Dot product = ax + by + cz.
• Example with vectors:
  • a = 2i + 3j + 4k, b = 7i + 5j + k.
  • Dot product = 2×7 + 3×5 + 4×1 = 33.

Finding the Angle Between Two Vectors

• Use the formula: cos(θ) = (a·b) / (|a|×|b|).
• θ is the angle between vectors a and b.
• Example:
  • a = 2i - 2j + k, b = 12i + 4j - 3k.
  • Dot product = 13.
  • Magnitude of a = √9 = 3.
  • Magnitude of b = √169 = 13.
  • cos(θ) = 13 / (3×13) = 1/3.
  • θ = cos⁻¹(1/3) ≈ 70°.

Nature of Vectors Based on Angle

• θ = 90°: Vectors are perpendicular (orthogonal).
• θ = 0°: Vectors are parallel.
• θ = 180°: Vectors are anti-parallel.

Properties of Dot Product

• Commutative Property:
  • a·b = b·a.
  • Example: V = 2i + 3j, U = 5i + 7j.
  • V·U = 31, U·V = 31.
• Orthogonality:
  • If a·b = 0, either vector is zero or they are perpendicular.
• Parallel Vectors:
  • Dot product = product of magnitudes.
  • θ = 0°, cos(0) = 1.
• Anti-parallel Vectors:
  • Angle θ = 180°, cos(180) = -1.
  • Dot product = - (product of magnitudes).