Difference between revisions of "Vector"

Revision as of 13:05, 29 November 2007

A vector is a magnitude with a direction. 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 $\vec{PQ}$ 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,i.e. length, and direction, vectors can be freely translated about a plane without changing them. Hence, it is convenient to consider a vector as originating from the origin. This way, two vectors can be compared only by 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, $(\vec{v}+\vec{w})^2=||\vec{v}||^2+||\vec{w}||^2-2||\vec{v}||||\vec{w}||\cos\theta$ .

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

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 $a,b$ ,

$\vec{x}+\vec{y}=\vec{y}+\vec{x}$ (Commutative in +)

$(\vec{x}+\vec{y})+\vec{z}=\vec{x}+(\vec{y}+\vec{z})$ (Associative in +)

There exists the zero vector $\vec{0}$ such that $\vec{x}+\vec{0}=\vec{x}$ (Additive identity)

For each $\vec{x}$ , there is a vector $\vec{y}$ such that $\vec{x}+\vec{y}=\vec{0}$ (Additive inverse)

$1\vec{x}=\vec{x}$ (Unit scalar identity)

$(ab)\vec{x}=a(b\vec{x})$ (Associative in scalar)

$a(\vec{x}+\vec{y})=a\vec{x}+a\vec{y}$ (Distributive on vectors)

$(a+b)\vec{x}=a\vec{x}+b\vec{x}$ (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 $\bold{u}\cdot\bold{v}=u_1v_1+u_2v_2+\cdots+u_nv_n$ .

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.

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

Related threads from AoPS forum

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 == Addition of Vectors ==
 For vectors <math>\vec{v}</math> and <math>\vec{w}</math>, with angle <math>\theta</math> formed by them, <math>(\vec{v}+\vec{w})^2=||\vec{v}||^2+||\vec{w}||^2-2||\vec{v}||||\vec{w}||\cos\theta</math>.
-***pictures would be helpful here***
+{{image}}
 From this it is simple to derive that for a real number <math>c</math>, <math>c\vec{v}</math> is the vector <math>\vec{v}</math> with magnitude multiplied by <math>c</math>.  Negative <math>c</math> corresponds to opposite directions.

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