Orthogonality & Orthonormal Sets

Dot product, norm, unit vectors, and orthonormal sets.

The idea

In $\mathbb{R}^{n}$ the dot product supplies both length and angle. For $u, v$ in $\mathbb{R}^{n}$ it is $u \cdot v = u_1 v_1 + \cdots + u_n v_n.$

Applied to a vector and itself, it gives $v \cdot v = v_1^{2} + \cdots + v_n^{2}$, which by the Pythagorean theorem is the squared length; the norm of $v$ is therefore $\lVert v \rVert = \sqrt{v \cdot v}$. Applied to two different vectors, its sign reports the angle between them: positive when they lean the same way, negative when they lean apart, and zero exactly when they meet at a right angle.

Definition (Orthogonal and orthonormal).

Two vectors $u$ and $v$ are orthogonal when $u \cdot v = 0$. A set of vectors is orthonormal when the vectors are mutually orthogonal and each has norm $1$.

A vector of norm $1$ is a unit vector, and dividing any nonzero $v$ by its norm produces one; this is normalizing $v$.

Ways to work on it

• Walkthrough. Dot product, orthogonality, norm, and the orthonormal definition.
• Practice. Test a pair of vectors for orthogonality.
• Hardest. Build a unit vector orthogonal to a given one to complete an orthonormal set.

Not sure where to start? Take the ten-question placement test.