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The Euclidean inner product is symmetric, linear in each argument, and positive-definite. Specifically, $\mathbf{a} \cdot \mathbf{a}$ equals the squared length of $\mathbf{a}$, which is positive for every nonzero vector $\mathbf{a}$.

Conditions: The vectors are real coordinate vectors.; The operation is the standard dot product.

The Euclidean inner product is symmetric, linear in each argument, and positive-definite. Specifically, $\mathbf{a} \cdot \mathbf{a}$ equals the squared length of $\mathbf{a}$, which is positive for every nonzero vector $\mathbf{a}$.

Conditions: The vectors are real coordinate vectors.; The operation is the standard dot product.

The dot product of a vector $\mathbf{a}$ with itself, $\mathbf{a} \cdot \mathbf{a}$, is equal to the square of its length (norm), $\|\mathbf{a}\|^2$.

Conditions: The vector is a real coordinate vector.; The length is the Euclidean norm.

The dot product of a vector $\mathbf{a}$ with itself, $\mathbf{a} \cdot \mathbf{a}$, is equal to the square of its length (norm), $\|\mathbf{a}\|^2$.

Conditions: The vector is a real coordinate vector.; The length is the Euclidean norm.