Vector

Revision as of 18:56, 12 February 2009 by Math154 (talk | contribs) (Dot (Scalar) Product)

The word vector has many different definitions, depending on who is defining it and in what context. Physicists will often refer to a vector as "a quantity with a direction and magnitude." For Euclidean geometers, a vector is essentially a directed line segment. In many situations, a vector is best considered as an n-tuple of numbers (often real or complex). Most generally, but also most abstractly, a vector is any object which is an element of a given vector space.

A vector is usually graphically represented as an arrow. Vectors can be uniquely described in many ways. The two most common is (for 2-dimensional vectors) by describing it with its length (or magnitude) and the angle it makes with some fixed line (usually the x-axis) or by describing it as an arrow beginning at the origin and ending at the pint $(x,y)$. An $n$-dimensional vector can be described in this coordinate form as an ordered $n$-tuple of numbers within angle brackets or parentheses, $(x\,\,y\,\,z\,\,...)$. The set of vectors over a field is called a vector space.

Description

Every vector $\overrightarrow{PQ}$ has a starting point $P\langle x_1, y_1\rangle$ and an endpoint $Q\langle x_2, y_2\rangle$. Since the only thing that distinguishes one vector from another is its magnitude or length, and direction, vectors can be freely translated about a plane without changing. Hence, it is convenient to consider a vector as originating from the origin. This way, two vectors can be compared by only looking at their endpoints. This is why we only require $n$ values for an $n$ dimensional vector written in the form $(x\,\,y\,\,z\,\,...)$. The magnitude of a vector, denoted $\|\vec{v}\|$, is found simply by using the distance formula.

Addition of Vectors

For vectors $\vec{v}$ and $\vec{w}$, with angle $\theta$ formed by them, $\|\vec{v}+\vec{w}\|^2=\|\vec{v}\|^2+\|\vec{w}\|^2+2\|\vec{v}\|\|\vec{w}\|\cos\theta$.

[asy]   size(150); pen p=linewidth(1); MA("\theta",(5,-1),(2,3),(4,6),0.3,9,yellow); MC("\vec v",D((0,0)--(2,3),orange+p,Arrow),NW); D((2,3)--(3,4.5)); MC("\vec w",D((2,3)--(5,-1),green+p,Arrow),NE); MC(-10,"\vec{v}+\vec{w}",D((0,0)--(5,-1),red+p,Arrow),S); [/asy]

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Addition of vectors

From this it is simple to derive that for a real number $c$, $c\vec{v}$ is the vector $\vec{v}$ with magnitude multiplied by $c$. Negative $c$ corresponds to opposite directions.

Properties of Vectors

Since a vector space is defined over a field $K$, it is logically inherent that vectors have the same properties as those elements in a field.

For any vectors $\vec{x}$, $\vec{y}$, $\vec{z}$, and real numbers $a,b$,

  1. $\vec{x}+\vec{y}=\vec{y}+\vec{x}$ (Commutative in +)
  2. $(\vec{x}+\vec{y})+\vec{z}=\vec{x}+(\vec{y}+\vec{z})$ (Associative in +)
  3. There exists the zero vector $\vec{0}$ such that $\vec{x}+\vec{0}=\vec{x}$ (Additive identity)
  4. For each $\vec{x}$, there is a vector $\vec{y}$ such that $\vec{x}+\vec{y}=\vec{0}$ (Additive inverse)
  5. $1\vec{x}=\vec{x}$ (Unit scalar identity)
  6. $(ab)\vec{x}=a(b\vec{x})$ (Associative in scalar)
  7. $a(\vec{x}+\vec{y})=a\vec{x}+a\vec{y}$ (Distributive on vectors)
  8. $(a+b)\vec{x}=a\vec{x}+b\vec{x}$ (Distributive on scalars)

Vector Operations

Dot (Scalar) Product

Consider two vectors $\bold{u}=\langle u_1,u_2,\ldots,u_n\rangle$ and $\bold{v}=\langle v_1, v_2,\ldots,v_n\rangle$ in $\mathbb{R}^n$. The dot product is defined as $\bold{u}\cdot\bold{v}=\|\bold{u}\|\|\bold{v}\|\cos\theta=u_1v_1+u_2v_2+\cdots+u_nv_n$, where $\theta$ is the angle formed by the two vectors.

Cross (Vector) Product

The cross product between two vectors $\bold{a}$ and $\bold{b}$ in $\mathbb{R}^3$ is defined as the vector whose length is equal to the area of the parallelogram spanned by $\bold{a}$ and $\bold{b}$ and whose direction is in accordance with the right-hand rule.

If $\vec{a}=\langle a_1,a_2,a_3\rangle$ and $\vec{b}=\langle b_1,b_2,b_3\rangle$, then the cross product of $\vec{a}$ and $\vec{b}$ is given by

$\vec{a}\times\vec{b}=\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\end{vmatrix}.$

where $\hat{i},\hat{j},\hat{k}$ are unit vectors along the co-ordinate axes.

Triple Scalar Product

The triple scalar product of three vectors $\bold{a,b,c}$ is defined as $(\bold{a}\times\bold{b})\cdot \bold{c}$. Geometrically, the triple scalar product gives the signed area of the parallelpiped determined by $\bold{a,b}$ and $\bold{c}$. It follows that

$(\bold{a}\times\bold{b})\cdot \bold{c} = (\bold{c}\times\bold{a})\cdot \bold{b} = (\bold{b}\times\bold{c})\cdot \bold{a}.$

It can also be shown that

$(\bold{a}\times\bold{b})\cdot \bold{c} = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}.$

Triple Vector Product

See Also

Discussion

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