Difference between revisions of "Vector"

Revision as of 22:09, 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 $P\langle x_1, y_1\rangle$ 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. 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 $\bold{u}=\langleu_1,u_2,\ldots,u_n\rangle$ (Error compiling LaTeX. Unknown error_msg) and $\bold{v}=\langlev_1, v_2,\ldots,v_n\rangle$ (Error compiling LaTeX. Unknown error_msg) 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 3: / Line 3: @@
 == Description ==
-Every vector <math>\vec{PQ}</math>has a starting point <math>P<x_1, y_1></math> and an endpoint <math>Q<x_2, y_2></math>.  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
+Every vector <math>\vec{PQ}</math>has a starting point <math>P\langle x_1, y_1\rangle</math> and an endpoint <math>Q\langle x_2, y_2\rangle</math>.  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 ==
@@ Line 18: / Line 18: @@
 == Vector Operations ==
 '''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}=\langleu_1,u_2,\ldots,u_n\rangle</math> and <math>\bold{v}=\langlev_1, v_2,\ldots,v_n\rangle</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>.

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