How do you find the angle between two nonzero vectors using the dot product?

To calculate the dot product, multiply corresponding components of the two vectors and add the products. The result is a scalar.

Conditions: The vectors must have the same number of components.; The vectors are real coordinate vectors.

Yes, the dot product is symmetric. This means that for any two vectors $\mathbf{a}$ and $\mathbf{b}$, $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$.

Conditions: The vectors are in a real inner product space.; 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.

Yes, the dot product distributes over vector addition. This means $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$.

Conditions: The vectors are in a real inner product space.; The operation is the standard dot product.

A zero dot product indicates that two vectors are orthogonal (form a right angle) if and only if both vectors are nonzero.

Conditions: Both vectors are nonzero.; The dot product of the two vectors is exactly zero.