6.1 - Inner Product, Length, and Orthogonality

For vectors:

,

The inner product is:

Properties

Length (Norm) of a Vector

OR:

Unit Vector (Normalization)

To make a unit vector in the direction of :

Example structure:

Distance Between Two Vectors

Distance between and :

Expanded:

Orthogonality

Vectors and are orthogonal if:

Orthogonal Perpendicular in

Pythagorean Theorem (Vector Version)
If

Parallelogram Law

For any vectors :

+

Finding a Unit Vector

Steps:

1. Compute

2. Multiply vector by /

Checking Orthogonality

Compute dot product

If result=0 then it is orthogonal

Distance between vectors

Subtract take norm