Orthogonality
When are two vectors orthogonal or not?
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Study Orthogonality
Definition 1. Two vectors \(\mathbf{v},\mathbf{w}\) in \(\mathbb{R}^2\) or \(\mathbb{R}^3\) are orthogonal if and only if \(\langle \mathbf{v},\mathbf{w} \rangle = 0\).
Example 2. Consider the vectors \[\mathbf{v}_1:=\begin{pmatrix} 1\\2 \end{pmatrix},\ \mathbf{v}_2:=\begin{pmatrix} 2\\-2 \end{pmatrix}\ \text{ and } \ \mathbf{v}_3:=\begin{pmatrix} -4\\2 \end{pmatrix}.\] Then \(\langle \mathbf{v}_1,\mathbf{v}_2\rangle = -2\) and \(\langle \mathbf{v}_1,\mathbf{v}_3 \rangle=0\). Hence, \(\mathbf{v}_1\) and \(\mathbf{v}_3\) are orthogonal. But \(\mathbf{v}_1\) and \(\mathbf{v}_2\) are not orthogonal.
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