What is the difference between orthogonal and orthonormal in terms of vectors and vector space?

I am beginner to linear algebra. I want to know detailed explanation of what is the difference between these two and geometrically how these two are interpreted?

Two vectors are orthogonal if their inner product is zero. In other words $\langle u,v\rangle =0$. They are orthonormal if they are orthogonal, but each vector has norm $1$. In other words $\langle u,v \rangle =0$ but $\langle u,u\rangle = \langle v,v\rangle =1$.

Example

For vectors in $\mathbb{R}^3$ let

$$u \;\; =\;\; \left[ \begin{array}{c} 1\\ 2\\ 0\\ \end{array} \right ] \hspace{2pc} v \;\; =\;\; \left [ \begin{array}{c} 0\\ 0\\ 3\\ \end{array} \right ].$$

The vectors $u$ and $v$ are orthogonal since

$$\langle u, v\rangle \;\; =\;\; 1\cdot 0 + 2\cdot 0 + 0\cdot 3 \;\; =\;\; 0$$

but they are not orthonormal since $||u|| = \sqrt{\langle u,u\rangle } = \sqrt{1 + 4} = \sqrt{5}$ and $||v|| = \sqrt{\langle v,v\rangle } = \sqrt{3^2} = 3$. If we define new vectors $\hat{u} = \frac{u}{||u||}$ and $\hat{v} = \frac{v}{||v||}$ then $\hat{u}$ and $\hat{v}$ are orthonormal since they each now have norm $1$, and orthogonality is preserved since $\langle \hat{u}, \hat{v}\rangle = \frac{\langle u,v\rangle }{||u||\cdot ||v||} = 0$.

You can think of orthogonality as vectors being perpendicular in a general vector space. And for orthonormality what we ask is that the vectors should be of length one. So vectors being orthogonal puts a restriction on the angle between the vectors whereas vectors being orthonormal puts restriction on both the angle between them as well as the length of those vectors. These properties are captured by the inner product on the vector space which occurs in the definition.

For example, in $\mathbb{R}^2$ the vectors $(0,2)$ and $(1,0)$ are orthogonal but not orthonormal because $(0,2)$ has length $2.$

• Does orthogonality of vectors means they are always perpendicular or any perpendicular vectors are orthogonal and rest are not ? What is mean by orthogonality of vector spaces? – Sanjeev Aug 4 '15 at 5:01
• I feel that orthogonality is a generalization of perpendicularity. For visual purposes or to get a feel for what exactly is happening you can think or vectors being orthogonal as the same as vectors being perpendicular, say in $\mathbb{R}^2$. But for an abstract vector space we don't define perpendicularity, we say vectors are orthogonal which have similar properties to perpendicular vectors in $\mathbb{R}^2$. – Makarand Sarnobat Aug 4 '15 at 5:15
• For the second question we can say when two vector spaces are orthogonal if the both are contained in an ambient space which is endowed with a inner product. In that case we say that two subspaces $V$ and $W$ are orthogonal if $<v,w> = 0$ for all $v \in V$ and $w \in W.$ – Makarand Sarnobat Aug 4 '15 at 5:19