Scalar product

Scalar Product

Overview

• The scalar product (also called the dot product) takes two vectors and returns a scalar (a single real number). This contrasts with the vector product which gives another vector as the result.

Computation

• The scalar product of two vectors a and b is calculated by multiplying together corresponding components of the vectors and then summing these products. If a = ai + bj + ck and b = xi + yj + zk, then the scalar product a . b = ax + by + cz.
• This method works for vectors of any dimension, as long as they have the same number of components.

Geometric Interpretation

• Geometrically, the scalar product of two vectors can be expressed as **a . b =

  a

  b

  cos θ, where **θ is the angle between the vectors and **

  a

  ** and **

  b

  ** are the magnitudes (lengths) of the vectors.

Angle between two vectors

• You can use the scalar product to find the angle between two vectors: rearrange the formula **a . b =

  a

  b

  cos θ** to solve for θ. The cos−1 function can be used to find the angle.

Perpendicular Vectors

• If two vectors are perpendicular (at a right angle to each other), their scalar product will be zero. This is because cos 90° = 0, and so the formula **a . b =

  a

  b

  cos 90°** becomes a . b = 0.

Projection

• The projection of one vector on another can be calculated using the scalar product. The projection of vector a onto vector b gives the length of the shadow that a would cast on b if the sun were shining directly overhead from the direction of a. The projection is given by **(a . b) /

  b

  **.

The scalar product plays a crucial role in many areas of mathematics, including geometry, physics, and engineering. Understanding its properties and how to compute it is a critical skill in these fields.