> [!Definition] Theorem > Let $n\geq 1$ and $\mathbb{R}^{n}$ denote the [[Real n-Space|real n-space]]. Let $\underline{v},\underline{w}\in\mathbb{R}^{n}$ with $\underline{w}\neq \underline{0}$ and let $\underline{\hat{w}}=\underline{w}/||w||$ be the [[Euclidean Norm#^746e28|unit vector]] in the direction of $\underline{w}$. > > The [[Dot Product in Real n-Space|scalar product]] $\underline{v}\cdot \underline{\hat{w}}$ is called the **component of $\underline{v}$ in the direction of $\underline{w}$**. > The vector $(\underline{v}\cdot \underline{\hat{w}})\underline{\hat{w}}$ is the **orthogonal projection of $\underline{v}$ in the direction of $\underline{w}$**. **Proof**: ![[orthogonal projection.png|400]] In the picture $\lambda$ must equal $v \cdot \underline{\hat{w}}$ since the dot product $\underline{n} \cdot \underline{\hat{w}}$ which must equal $0$ for the vectors be orthogonal.