Scalar Product of Vectors

Scalar Product of Vectors

• The scalar product or dot product of two vectors A and B is a scalar quantity.
• Defined as: A.B = AB cos θ, where θ is the angle between the vectors A and B.
• Geometric interpretation: A.B is the product of the magnitude of A and the component of B along A.
• Alternatively: It's the product of the magnitude of B and the component of A along B.
• Commutative law: A.B = B.A
• Distributive law: A.(B + C) = A.B + A.C
• Scalar multiplication: λA.(λB) = λ(A.B), where λ is a real number.
• For unit vectors i, j, k:
  • i.i = j.j = k.k = 1
  • i.j = j.k = k.i = 0
• Given two vectors A = Ax i + Ay j + Az k and B = Bx i + By j + Bz k, their scalar product is: A.B = AxBx + AyBy + AzBz
• Special cases:
  • A.A = A² = Ax² + Ay² + Az²
  • A.B = 0 if A and B are perpendicular (θ = 90°).

Example: Finding Angle and Projection

• Calculate the angle between force F = (3i + 4j - 5k) and displacement d = (5i + 4j + 3k).
  • Calculate F.d = 16
  • Calculate F.F = 50
  • Calculate d.d = 50
  • Use the formula: cos θ = F.d / (|F| |d|) = 16 / 50 = 0.32
  • Find θ = cos⁻¹(0.32)
• Projection of F on d:
  • The projection of F onto d is F.d / |d| = 16 / √(50) units.