Difference between revisions of "Vector"

Revision as of 20:20, 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, $|(\vec{v}+\vec{w})|^2=|\vec{v}|^2+|\vec{w}|^2-2|\vec{v}||\vec{w}|\cos\theta$ .

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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.

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

Discussion

This is a thread about what vectors are.

@@ 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 ==
-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.
@@ Line 18: / Line 19: @@
 #<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 +)
 #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]])
 #<math>1\vec{x}=\vec{x}</math> (Unit scalar identity)
 #<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+b)\vec{x}=a\vec{x}+b\vec{x}</math> ([[Distributive]] on scalars)
@@ Line 36: / Line 30: @@
 '''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>.
 '''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
 <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
 <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
@@ Line 61: / Line 52: @@
 *[[Matrix]]
 *[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.]
-{{stub}}

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