# Vector **A quantity or geometric object that possesses both a magnitude and direction.** ![[Vector.svg|400]] *A Euclidean vector* A **vector** originally referred to quantities which can only be properly described and expressed by both *magnitude* and *direction*. This is in contrast to a scalar which can be described by a single number, its magnitude. A *Euclidean vector* is a geometric object that also has both magnitude and direction and are formally *directed line segments*, or arrows, in Euclidean space. In most contexts, they are also simply referred to as vectors as they can represent vector quantities. Euclidean vectors can be thought of as the geometric object that brings some starting point to some terminal point. Euclidean vectors are often generalised into [[Vector Space|vector spaces]] whose elements are vector quantities which can be added together and scaled by scalars. Vectors are also assigned to every point of Euclidean space to create a [[vector field]], a common model in physics. The notation of the [[derivative]] and [[integral]] are extendable to vector fields which gives rise to vector calculus. [[Pseudovector|Pseudovectors]] are vector-like quantities which transform similarly but not exactly like vectors and are used to describe some physical quantities. > [!NOTE] Unit vector > A *unit vector* is a vector with magnitude one, typically used to represent a direction in vector operations. > > Any vector can be reduced to a unit vector with the same direction by the process of [[Vector#Normalisation|normalisation]]. A unit vector in the same direction as some vector $\mathbf{v}$ is typically denoted as $\hat{\mathbf{v}}$. ^ed5220 > [!NOTE] Zero vector > The *zero vector* is a vector with magnitude zero, denoted either as > $\mathbf{0}, \vec{0}, 0$ > It is the only vector which has no defined direction. ## Notation Vectors are typically notated with lowercase letters either in bold, $\mathbf{a}$, or italicised with an arrow above it, $\vec{a}$. If the vector $\mathbf{a}$ is directed from the point $A$ to the point $B$ then it can be notated as $\overrightarrow{AB}$ instead. ### Euclidean vector For a vector $\mathbf{a}$ directed from point $A$ to point $B$, point $A$ is known as its *origin*, *tail*, or *initial point* while point $B$ is known as its *head*, *tip*, or *terminal or final point*. On a two-dimensional diagram, a vector that points perpendicularly into or out of the diagram are represented by a circle with a cross $\otimes$ or a dot $\odot$ in the center, respectively. A general vector in $n$-dimensional Euclidean space, or $\mathbb{R}^{n}$, can be notated as a list of the [[Cartesian Coordinate System|Cartesian coordinates]] of its tip. These coordinates are also known as the *scalar components* of the vector and derive from the [[scalar projection]] of the vector onto the coordinate axes. $\mathbf{a}=(a_{1},a_{2},\dots,a_{n})$ ![[VectorScalarComponents.svg|350]] The scalar components of a vector are most commonly arranged into a *column* or *row vector*. A row vector is typically used in operations with [[Matrix|matrices]]. Both square brackets and parentheses are commonly used to surround vectors. $\mathbf{a}=\left[\begin{matrix}a_{1}\\ a_{2}\\\vdots\\ a_{n}\end{matrix}\right]=\left[\begin{matrix}a_{1}&a_{2}&\cdots&a_{n}\end{matrix}\right]^{\text{T}}$ The scalar components of a vector can also be used with the [[Basis#Standard basis|standard basis]] vectors to denote it as the resultant vector of a vector sum. $\mathbf{a}=a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}+\cdots+a_{n}\mathbf{e}_{n}$ In $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$, these standard basis vectors may also be denoted as $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$, respectively, for the $x$, $y$, and $z$ axes. ![[VectorScalarComponentsStandardBasis.svg|350]] ## Operations In the following vector operation definitions the vectors $\mathbf{a}$ and $\mathbf{b}$ are defined in the same Cartesian coordinate system with the origin as their initial point, where $\mathbf{a}=(a_{1}, a_{2},\dots,a_{n})\quad\mathbf{b}=(b_{1},b_{2},\dots,b_{n})$ All of the following operations are generalisable to any dimension, except the cross product. ### Addition > [!NOTE] Addition > $\mathbf{a}+\mathbf{b}=(a_{1}+b_{1}, a_{2}+b_{2},\dots,a_{n}+b_{n})$ The resulting vector of the sum of vectors is often known as the *resultant vector* of the operand vectors. Graphically, the vector addition of $\mathbf{a}$ and $\mathbf{b}$ can be represented by placing the tail of $\mathbf{b}$ at the head of $\mathbf{a}$. The resultant vector $\mathbf{a}+\mathbf{b}$ is the vector from the tail of $\mathbf{a}$ to the head of $\mathbf{b}$. An alternate representation is to form a parallelogram by placing the tails $\mathbf{a}$ and $\mathbf{b}$ at the same point, in which case $\mathbf{a+b}$ is the diagonal of the parallelogram from the tails. ![[VectorAddition.svg]] ### Subtraction > [!NOTE] Subtraction > $\mathbf{a}-\mathbf{b}=(a_{1}-b_{1}, a_{2}-b_{2},\dots,a_{n}-b_{n})$ As in vector addition, the resulting vector is also often known as the resultant vector. The vector subtraction of $\mathbf{b}$ from $\mathbf{a}$ can be graphically represented by placing the tail of $-\mathbf{b}$, the reverse vector of $\mathbf{b}$, at the head of the $\mathbf{a}$. The resultant vector $\mathbf{a}-\mathbf{b}$ is the vector from the tail of $\mathbf{a}$ to the head of $\mathbf{-b}$. Alternatively, if the tails of $\mathbf{a}$ and $\mathbf{b}$ are placed at the same point then the resultant vector is the vector from the head of $\mathbf{b}$ to the head of $\mathbf{a}$. This can be seen as equivalent to the representation of the vector addition $\mathbf{b}+(\mathbf{a}-\mathbf{b})=\mathbf{a}$ and is the diagonal of the parallelogram formed from $\mathbf{a}$ and $\mathbf{b}$ from the head of $\mathbf{b}$. ![[VectorSubtraction.svg]] ### Scalar multiplication > [!NOTE] Scalar multiplication > $k\mathbf{a}=(ka_{1},ka_{2},\dots, ka_{n})$ The scalar multiplication of a vector is also known as *scaling* the vector. Graphically, a vector multiplied by a scalar $k$ is stretched by a factor of $k$ along its original direction. An alternative representation is the resulting vector after the vector is added end-to-end $k$ times. If the scalar is negative, then the direction of the vector is reversed. ![[VectorScalarMultiplication.svg|450]] ### Magnitude > [!NOTE] Magnitude > The *magnitude*, *length*, or *norm* of a vector $\mathbf{a}$ is > $\|\mathbf{a}\|=\sqrt{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}$ The formula for the magnitude of a vector derives directly from the application of the Pythagorean theorem to find Euclidean distance. ![[VectorMagnitude.svg|350]] #### Normalisation > [!NOTE] Normalisation > $\hat{\mathbf{a}}=\frac{\mathbf{a}}{\|\mathbf{a}\|}=\left(\frac{a_{1}}{\|\mathbf{a}\|},\frac{a_{2}}{\|\mathbf{a}\|},\cdots,\frac{a_{n}}{\|\mathbf{a}\|}\right)$ The normalisation of a vector gives a unit vector which points in the same direction as the vector. ### Product There are two notions of the product of two vectors, the *[[dot product]]* and the *[[cross product]]*. The dot product is defined in any $n$-dimension Euclidean space, whereas the cross product is defined in only three dimension Euclidean space. ![[CrossProduct.svg|300]] *Geometric representation of the cross product*