> [!NOTE] Lemma > Let $\mathbb{R}^{n}$ denote a [[Real n-Space|real n-Space]]. Let $\underline{u},\underline{v}\in \mathbb{R}^{n} \setminus \{ \underline{0} \}.$ Let $\theta \in [0,\pi]$ denote the [[Angle Between Nonzero Real Vectors|acute angle]] between $\underline{u}$ and $\underline{v}.$ Then their [[Dot Product in Real n-Space|dot product]] is given by $\underline{u}\cdot \underline{v} = ||\underline{u}|| \cdot ||\underline{v}||\cdot \cos \theta$where $||\dots||$denotes [[Euclidean Norm|length]] **Proof**: By definition, $\theta = \cos^{-1}\left( \frac{\underline{u} \cdot \underline{v}}{||\underline{u}|| \cdot || \underline{v}||} \right)$so taking $\cos$ of both sides gives the required result.