> [!Definition] > Given a field, $\mathbb{F}$, an $n$-dimensional vector over $\mathbb{F}$ is any element of $\mathbb{F}^{n}$ (which is a [[Vector Space|vector space]] over $\mathbb{F}$). > > [!Example] Example (Euclidean Vector) > A vector over the reals is known as a [[Euclidean Vector|Euclidean vector]]. # Properties Vector Algebra: We can define the following binary operation and binary function [[Vector Addition, Subtraction & Scalar Multiplication]] respectively. Two vectors are [[Collinearity of Two Vectors|collinear]] if they have the same direction. See [[Span of Subset of Vector Space]]; [[Linear Independence]]; # Applications - [[Vector Field on Subset of Real n-Space]].