Cauchy–Schwarz inequality for vectors, with a worked example

The dot product and the lengths of two vectors are connected. Cauchy–Schwarz tells us how large the absolute dot product can be. We can also use the equality case to tell whether the vectors are linearly dependent.

Compare the absolute dot product with the lengths

| u \cdot v | \leq ‖ u ‖ ‖ v ‖

The inequality holds for vectors of the same dimension. Equality holds exactly when the pair is linearly dependent, including when one of the vectors is zero.

Find both sides for two vectors

Let $u = (3, 4)$ and $v = (- 6, - 8)$ . First, find the absolute value of the dot product.

| u \cdot v | = | 3 (- 6) + 4 (- 8) | = 50

Then find the lengths and multiply them.

\begin{array}{l} ‖ u ‖ = \sqrt{3^{2} + 4^{2}} = 5 \\ ‖ v ‖ = \sqrt{(- 6)^{2} + (- 8)^{2}} = 10 \\ ‖ u ‖ ‖ v ‖ = 50 \end{array}

Use equality to identify dependence

Both sides equal $50$ , so this pair is linearly dependent. We can also see the relationship directly because $v = - 2 u$ .

For nonzero vectors, a strict inequality means that neither vector is a scalar multiple of the other. That pair is linearly independent.

Learn mathKrista KingAugust 1, 2021math, learn online, online course, online math

Cauchy–Schwarz inequality for vectors, with a worked example

Compare the absolute dot product with the lengths

Find both sides for two vectors

Use equality to identify dependence

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