**Definition:** Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are *orthogonal* if their inner product (dot product) is zero:
\[\mathbf{u} \perp \mathbf{v} \iff \langle \mathbf{u}, \mathbf{v} \rangle = 0\]

In \(\mathbb{R}^n\) with the standard dot product:
\[\mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^n u_i v_i = 0\]

**Orthogonal projection:** The projection of \(\mathbf{v}\) onto a nonzero vector \(\mathbf{u}\) is:
\[\text{proj}_{\mathbf{u}}(\mathbf{v}) = \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}\]

This gives the closest point on the line spanned by \(\mathbf{u}\) to \(\mathbf{v}\). The error \(\mathbf{v} - \text{proj}_{\mathbf{u}}(\mathbf{v})\) is orthogonal to \(\mathbf{u}\).

**Example:** The projection of \((3,4)\) onto \((1,0)\) is \((3,0)\). The error \((0,4)\) is orthogonal to \((1,0)\).

**Orthogonal decomposition:** Any vector \(\mathbf{v}\) can be written as \(\mathbf{v} = \mathbf{v}_W + \mathbf{v}_{W^\perp}\) where \(\mathbf{v}_W\) lies in a subspace \(W\) and \(\mathbf{v}_{W^\perp}\) is orthogonal to every vector in \(W\).