How do you calculate the dot product of two vectors from their components?

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.

Divide the dot product of the two vectors by the product of their lengths to obtain $\cos \theta$. Then, take the arccosine of this value to find the angle $\theta$ in the range $[0, \pi]$.

Conditions: Both vectors are nonzero.; The vectors are in a real inner product space (e.g., $\mathbb{R}^n$).

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.