Scalar product. Relationships

The scalar product of two vectors $\vec{a}$ and $\vec{b}$ is defined as $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$, where $\theta$ is the angle between them. This product is useful for finding angles, checking for perpendicularity, and projecting one vector onto another.

Important properties

• If $\vec{a} \cdot \vec{b} = 0$, then $\vec{a}$ and $\vec{b}$ are perpendicular.

• The scalar product is commutative: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$.

• It is distributive over addition: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$.

• The scalar product can be used to find the angle between two vectors: $\cos \theta = \dfrac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}$.