Difference between revisions of "Vector"

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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 <math>(x,y)</math>. An <math>\displaystyle n</math>-dimensional vector can be described in this coordinate form as an ordered <math>\displaystyle n</math>-tuple of numbers within angle brackets or parentheses, <math>(x\,\,y\,\,z\,\,...)</math>. The set of vectors in some space is an example of a [[vector space]].
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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 <math>(x,y)</math>. An <math>n</math>-dimensional vector can be described in this coordinate form as an ordered <math>n</math>-tuple of numbers within angle brackets or parentheses, <math>(x\,\,y\,\,z\,\,...)</math>. The set of vectors over a [[field]] is called a [[vector space]].
  
  
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Form 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.
 
Form 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.
 
== Properties of Vectors ==
 
== Properties of Vectors ==
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Since a [[vector space]] is defined over a [[field]] <math>K</math>, it is logically inherent that vectors have the same properties as field properties.
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For vectors <math>\vec{v}</math> and <math>\vec{w}</math>,  
 
For vectors <math>\vec{v}</math> and <math>\vec{w}</math>,  
  

Revision as of 10:44, 15 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, $(\vec{v}+\vec{w})^2=||\vec{v}||^2+||\vec{w}||^2-2||\vec{v}||||\vec{w}||\cos\theta$.

      • pictures would be helpful here***

Form 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 field properties.

For vectors $\vec{v}$ and $\vec{w}$,

(i)

(ii)

(iii)

(iv)

...

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 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

See Also

Related threads from AoPS forum


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