Use of dot product to check perpendicularity of two vectors

Two nonzero vectors a and b are called perpendicular or orthogonal if the angle between them is \(\theta = \pi/2\). Then Theorem 3 gives \[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos(\pi/2) = 0 \] and conversely if \(\mathbf{a} \cdot \mathbf{b} = 0\), then \(\cos\theta = 0\), so \(\theta = \pi/2\). The zero vector 0 is considered to be perpendicular to all vectors. Therefore we have the following method for determining whether two vectors are orthogonal.

Theorem 7: Two vectors a and b are orthogonal if and only if

\[\mathbf{a} \cdot \mathbf{b} = 0\].

EXAMPLE 4 Show that \(2\mathbf{i} + 2\mathbf{j} - \mathbf{k}\) is perpendicular to \(5\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}\).