Scalar Products

Scalar Products - Definition and Basic Principles

• Scalar products, also known as dot products, are an operation between two vectors that results in a scalar quantity.
• The scalar product of two vectors a and b is denoted as a.b.
• When two vectors are perpendicular, their scalar product is zero because the cosine of the angle between them is zero.

• The scalar product of a.a (a vector with itself) equals the square of its magnitude, i.e., **

  a

  ^2**.

Calculation of Scalar Products

• The scalar product a.b can be calculated geometrically as **

  a

  *

  b

  * cos θ**, where θ is the angle between vectors a and b.

• However, in Cartesian coordinates, the scalar product of a = ai + bj + ck and b = di + ej + fk is given by a.d + b.e + c.f.
• Be aware of the potential need to convert between Cartesian and geometric representations to calculate scalar products.

Application of Scalar Products

• Scalar products have multiple applications, including determining the angle between two vectors.

• **θ = cos^-1 ((a.b)/(

  a

  b

  ))** gives the angle between vectors a and b, where a.b is the scalar product of a and b and **

  a

  ,

  b

  ** are the magnitudes (lengths) of vectors a and b respectively.

• Scalar products are also useful in projections: the projection of vector a on b equals **(a.b)/

  b

  **.

Properties of Scalar Products

• Scalar products have several important properties. They are commutative (a.b = b.a), distributive (a.(b+c) = a.b + a.c) but not associative.
• Unlike the cross product, scalar products have no role in producing a vector that is perpendicular to the given vectors. Its main utility lies in producing scalar quantities and the discussion of angles between vectors.