Vector Scalar Product

Definitions
Orthogonality

Definitions

The scalar product (or, inner product, or dot product) between two vectors $x,y in mathbf{R}^n$ is the scalar denoted x^Ty , and defined as

$x^Ty = sum_{i=1}^n x_i y_i.$

The motivation for our notation above will come later, when we define the matrix-vector product. The scalar product is also sometimes denoted langle x, y rangle , a notation which originates in physics.

Matlab syntax
>> x = [1; 2; 3]; y = [4; 5; 6];
>> scal_prod = x'*y;

Examples:

Rate of return of a financial portfolio.
Sample and weighted average.
Beer-Lambert law in absorption spectroscopy.

Orthogonality

We say that two vectors $x,y in mathbf{R}^n$ are orthogonal if x^Ty = 0 .

Example: Two orthogonal vectors in $mathbf{R}^3$ .