Orthogonal and Orthonormal Vectors

Orthogonal Vectors

Two vectors are orthogonal or perpendicular if their dot product is zero.

Orthogonal Vectors

Example

Orthogonal Vector Example

Orthogonal Vector Solution

Not perpendicular.


Orthonormal Vectors

Two vectors are orthonormal if:

1. Their dot product is zero.

2.The two vectors are unit vectors.


Unit Vectors

Perpendicular Vectors

Unit Vectors

Unit Vectors

Unit Vectors

Examples

Calculate the value of k for the vectors Vector = (1, k) and = (−4, k) knowing that they are orthogonal.

Vector · = 0 −4 + m2 = 0; m = ± 2


If { Vector, Vector} is an orthonormal basis, calculate:

1 Vector · Vector = 1 · 1 · cos 0° = 1

2 Vector · Vector = 1 · 1 · cos 90° = 0

3 Vector · Vector = 1 · 1 · cos 90° = 0

4 Vector · Vector = 1 · 1 · cos 0° = 1


If {Vector, Vector} is an orthonormal basis and Vectors are:

Orthonormal Vectors

Calculate the value of k knowing that Vector Solution.

Vector Calculations

Vector Calculations

Vector Calculations

Vector Solution


If {Vector, Vector} is an orthonormal basis and Vectors are:

Orthogonal Vector Example

Calculate the value of k for the two orthogonal vectors.

Orthogonal Vector Calculations

Orthogonal Vector Calculations

Orthogonal Vector Solution