Difference between revisions of "Vector"

Revision as of 21:07, 30 September 2006

A vector is a magnitude with a direction. Much of physics deals with vectors. An $\displaystyle n$ -dimensional vector can be thought of as an ordered $\displaystyle n$ -tuple of numbers within angle brackets. The set of vectors in some space is an example of a vector space.

Description

Every vector $\vec{PQ}$ has a starting point $P<x_1, y_1>$ and an endpoint $Q<x_2, y_2>$ . 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. The magnitude of a vector, denoted is found simply by using the distance formula.

Properties of Vectors

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Vector Operations

Dot (Scalar) Product Consider two vectors $\bold{u}=<u_1,u_2,\ldots,u_n>$ and $\bold{v}=<v_1, v_2,\ldots,v_n>$ in $\mathbb{R}^n$ . The dot product is defined as $\bold{u}\cdot\bold{v}=u_1v_1+u_2v_2+...+u_nv_n$ .

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 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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@@ Line 17: / Line 17: @@
 == Vector Operations ==
-'''Dot (Scalar) Product''' (proof as well? ) (--use law of cosines; I'm not good at proofs--from Aryth)
+'''Dot (Scalar) Product'''
+Consider two vectors <math>\bold{u}=<u_1,u_2,\ldots,u_n></math> and <math>\bold{v}=<v_1, v_2,\ldots,v_n></math> in <math>\mathbb{R}^n</math>.  The dot product is defined as <math>\bold{u}\cdot\bold{v}=u_1v_1+u_2v_2+...+u_nv_n</math>.
-Consider two vectors <math>\bold{u}=<u_1,u_2,...,u_n></math> and <math>\bold{v}=<v_1, v_2,...,v_n></math>.  The dot product is defined as <math>\bold{u}\cdot\bold{v}=u_1v_1+u_2v_2+...+u_nv_n</math>.
-In two or three dimensions, the dot product has the special geometric property that <math>\cos{\theta}=\frac{\bold{u}\cdot\bold{v}}{\|\bold{u}\|\|\bold{v}\|}</math>
+'''Cross (Vector) Product'''
+The cross product between two vectors <math>\bold{a}</math> and <math>\bold{b}</math> in <math>\mathbb{R}^3</math> is defined as the vector whose length is equal to the area of the parallelogram spanned by <math>\bold{a}</math> and <math>\bold{b}</math> and whose direction in accordance with the [[right-hand rule]].
-'''Cross (Vector) Product'''
+'''Triple Scalar product'''  The triple scalar product of three vectors <math>\bold{a,b,c}</math> is defined as <math>(\bold{a}\times\bold{b})\cdot \bold{c}</math>.  Geometrically, the triple scalar product gives the signed area of the parallelpiped determined by <math>\bold{a,b}</math> and <math>\bold{c}</math>.  It follows that
+<center><math>(\bold{a}\times\bold{b})\cdot \bold{c} = (\bold{c}\times\bold{a})\cdot \bold{b} = (\bold{b}\times\bold{c})\cdot \bold{a}.</math></center>
-'''Triple Scalar product'''
+It can also be shown that
+<center><math>(\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}.</math></center>
 '''Triple Vector Product'''

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