Difference between revisions of "Vector"

(Vector Operations)
m (fixes)
Line 1: Line 1:
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 <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]].
+
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 <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]].
  
 
== Description ==
 
== Description ==
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. This is why we only require <math>n</math> values for an <math>n</math> dimensional vector written in the form <math>(x\,\,y\,\,z\,\,...)</math>. The magnitude of a vector, denoted <math>||\vec{v}||</math>, 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. This is why we only require <math>n</math> values for an <math>n</math> dimensional vector written in the form <math>(x\,\,y\,\,z\,\,...)</math>. The magnitude of a vector, denoted <math>||\vec{v}||</math>, is found simply by  
 
using the distance formula.
 
using the distance formula.
  
Line 18: Line 19:
  
 
#<math>\vec{x}+\vec{y}=\vec{y}+\vec{x}</math> ([[Commutative]] in +)
 
#<math>\vec{x}+\vec{y}=\vec{y}+\vec{x}</math> ([[Commutative]] in +)
 
 
#<math>(\vec{x}+\vec{y})+\vec{z}=\vec{x}+(\vec{y}+\vec{z})</math> ([[Associative]] in +)
 
#<math>(\vec{x}+\vec{y})+\vec{z}=\vec{x}+(\vec{y}+\vec{z})</math> ([[Associative]] in +)
 
 
#There exists the zero vector <math>\vec{0}</math> such that <math>\vec{x}+\vec{0}=\vec{x}</math> ([[Additive identity]])
 
#There exists the zero vector <math>\vec{0}</math> such that <math>\vec{x}+\vec{0}=\vec{x}</math> ([[Additive identity]])
 
 
#For each <math>\vec{x}</math>, there is a vector <math>\vec{y}</math> such that <math>\vec{x}+\vec{y}=\vec{0}</math> ([[Additive inverse]])
 
#For each <math>\vec{x}</math>, there is a vector <math>\vec{y}</math> such that <math>\vec{x}+\vec{y}=\vec{0}</math> ([[Additive inverse]])
 
 
#<math>1\vec{x}=\vec{x}</math> (Unit scalar identity)
 
#<math>1\vec{x}=\vec{x}</math> (Unit scalar identity)
 
 
#<math>(ab)\vec{x}=a(b\vec{x})</math> ([[Associative]] in scalar)
 
#<math>(ab)\vec{x}=a(b\vec{x})</math> ([[Associative]] in scalar)
 
 
#<math>a(\vec{x}+\vec{y})=a\vec{x}+a\vec{y}</math> ([[Distributive]] on vectors)
 
#<math>a(\vec{x}+\vec{y})=a\vec{x}+a\vec{y}</math> ([[Distributive]] on vectors)
 
 
#<math>(a+b)\vec{x}=a\vec{x}+b\vec{x}</math> ([[Distributive]] on scalars)
 
#<math>(a+b)\vec{x}=a\vec{x}+b\vec{x}</math> ([[Distributive]] on scalars)
  
Line 36: Line 30:
 
'''Dot (Scalar) Product'''  
 
'''Dot (Scalar) Product'''  
 
Consider two vectors <math>\bold{u}=\langle u_1,u_2,\ldots,u_n\rangle</math> and <math>\bold{v}=\langle v_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+\cdots+u_nv_n</math>.
 
Consider two vectors <math>\bold{u}=\langle u_1,u_2,\ldots,u_n\rangle</math> and <math>\bold{v}=\langle v_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+\cdots+u_nv_n</math>.
 
  
 
'''Cross (Vector) Product'''
 
'''Cross (Vector) Product'''
Line 43: Line 36:
 
If <math>\vec{a}=\langle a_1,a_2,a_3\rangle</math> and <math>\vec{b}=\langle b_1,b_2,b_3\rangle</math>, then the cross product of <math>\vec{a}</math> and <math>\vec{b}</math> is given by  
 
If <math>\vec{a}=\langle a_1,a_2,a_3\rangle</math> and <math>\vec{b}=\langle b_1,b_2,b_3\rangle</math>, then the cross product of <math>\vec{a}</math> and <math>\vec{b}</math> is given by  
 
<center><math>\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}.</math></center>
 
<center><math>\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}.</math></center>
where <math>\hat{i},\hat{j},\hat{k}</math> are [[Unit vector|unit vectors]] along the co-ordinate axes.
+
where <math>\hat{i},\hat{j},\hat{k}</math> are [[unit vector]]s along the co-ordinate axes.
 
 
  
 
'''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  
 
'''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>
 
<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>
 
  
 
It can also be shown that  
 
It can also be shown that  
Line 61: Line 52:
 
*[[Matrix]]
 
*[[Matrix]]
 
*[http://www.artofproblemsolving.com/Forum/index.php?f=346\ Matrix-Linear Algebra AOPS forum]
 
*[http://www.artofproblemsolving.com/Forum/index.php?f=346\ Matrix-Linear Algebra AOPS forum]
== Related threads from AoPS forum ==
 
  
 +
== Discussion ==
 
*[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=89911\ This is a thread about what vectors are.]
 
*[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=89911\ This is a thread about what vectors are.]
 
 
{{stub}}
 

Revision as of 19:20, 4 February 2008

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 $\vec{PQ}$ 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 $a,b$,

  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 $\bold{u}\cdot\bold{v}=u_1v_1+u_2v_2+\cdots+u_nv_n$.

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.

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

Invalid username
Login to AoPS