Scalar product

Introduction to Scalar Product

  • The scalar product, also known as the dot product, is a fundamental operation in vector algebra. It takes two vectors and produces a scalar (a single real number).

  • The scalar product of two vectors a and b is denoted as a.b or a • b.

  • The scalar product is commutative, meaning a.b = b.a.

  • The scalar product is distributive: a.(b+c) = a.b + a.c.

Calculating the Scalar Product

  • The scalar product of two vectors can be calculated in two ways: magnitude and direction approach, or componentwise approach.

  • In magnitude and direction approach, the dot product is calculated as **a.b = a   b cosθ**, where θ is the angle between the vectors, a and b are their magnitudes.
  • The result of the scalar product is positive if 0 ≤ θ < 90°, zero if θ = 90°, and negative if 90 < θ ≤ 180°.

  • Using the componentwise approach, in 3D space, if a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), then a.b is calculated as a.b = a₁b₁ + a₂b₂ + a₃b₃.

Scalar Product and Orthogonal Vectors

  • Two vectors are orthogonal (perpendicular) to each other if their scalar product is zero. This implies either one or both vectors are zero, or they are at 90 degrees to each other.

Applications of the Scalar Product

  • The scalar product has a wide range of applications including determining the angle between two vectors, testing for orthogonality, projecting one vector onto another, and calculating work done by a force in physics.

  • The projection of vector a onto vector b (proj_b(a)) can be found by multiplying the unit vector of b (b/ b ), with the scalar product of a and b (a.b): **proj_b(a) = ((a.b)/ b )(b/ b ) = (a.b) b **.
  • In physics, the work W done by a constant force F acting along the line of motion is given as the scalar product of F and the displacement vector s: W = F.s.