Difference between revisions of "Vector"

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'''Vectors ''' are quantities that are specified by both a direction and magnitude of length. The way you draw a vector is by drawing an arrow. This arrow has a length and points in some direction. [[Image: vectora.jpg|right|150px]]
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'''Vectors ''' are quantities that are specified by both a direction and magnitude of length. The way you draw a vector is by drawing an arrow. This arrow has a length and points in some direction. [[Image: vectora.jpg|right|160px]]
  
  
When we use vectors we need to introduce the coordinate system. We represent vectors in terms of their components along the coordinate axes. So if we have an arrow we can now communicate with someone else about how much the vector goes in the x direction, the y direction and the z direction by using unit vectors. A vector is a combination of unit vectors that are placed along the the coordinate axes. These unit vector have length one. A unit vector is denoted by a lowercase letter with a hat above it, like so, <math>\hat{a}</math>. It is pronounced as a-hat.  Our standard correspond to the x-y-z coordinate system. <math>\hat{i}</math> points along the x-axis <math>\hat{j}</math> points along the y-axis and <math>\hat{k}</math> points along the z-axis. We can express any vector in terms of its components relying on the unit vectors. This helps us understand the direction. As for the magnitude, the length of a vector, we can just measure its length. The magnitude is denoted by <math>\left| \vec A \right|</math>. The magnitude of a vector is a '''scalar''' quantity, a numerical value with no direction.  [[Image: unitvec.jpg|left|200px]]
 
  
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:When we use vectors we need to introduce the coordinate system. We represent vectors in terms of their components along the coordinate axes. So if we have an arrow we can now communicate with someone else about how much the vector goes in the x direction, the y direction and the z direction by using unit vectors. A vector is a combination of unit vectors that are placed along the the coordinate axes. These unit vector have length one. A unit vector is denoted by a lowercase letter with a hat above it, like so, <math>\hat{a}</math>. It is pronounced as a-hat.  Our standard correspond to the x-y-z coordinate system. <math>\hat{i}</math> points along the x-axis <math>\hat{j}</math> points along the y-axis and <math>\hat{k}</math> points along the z-axis. We can express any vector in terms of its components relying on the unit vectors. This helps us understand the direction. As for the magnitude, the length of a vector, we can just measure its length. The magnitude is denoted by <math>\left| \vec A \right|</math>. The magnitude of a vector is a '''scalar''' quantity, a numerical value with no direction.  [[Image: unitvec.jpg|left|200px]]
  
When we have a vector quantity we put an arrow on top of the labeling letter to remind us that it is a vector. It looks like so <math>\vec A</math>. With <math>\left \langle a_1, a_2, a_3 \right \rangle</math> as any numerical values we can write any vector in the following form <math>\vec A = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}</math>. Another notation for a vector is <math> \vec A = \left \langle a_1, a_2, a_3 \right \rangle</math>. The components of a vector can be are written as: <math> \begin{bmatrix} 3 \\ 2\\ \end{bmatrix} </math> the vector has an x-component of 3 and a y-component of 2. [[Image:Vector bold1.PNG|thumb|The Euclidean vector (3, 2): Click to enlarge|250px|right]]  If we are given <math>\vec A = 3 \hat{i} + 3 \hat{j} + 2 \hat{k}</math> we know that vector A is 3 units in the x and y direction and 2 units in the z direction. [[Image: VecA.jpg|left|200px]]  
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:When we have a vector quantity we put an arrow on top of the labeling letter to remind us that it is a vector. It looks like so <math>\vec A</math>. With <math>\left \langle a_1, a_2, a_3 \right \rangle</math> as any numerical values we can write any vector in the following form <math>\vec A = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}</math>. Another notation for a vector is <math> \vec A = \left \langle a_1, a_2, a_3 \right \rangle</math>. The components of a vector can be are written as: <math> \begin{bmatrix} 3 \\ 2\\ \end{bmatrix} </math> the vector has an x-component of 3 and a y-component of 2.  
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[[Image:Vector bold1.PNG|thumb|The Euclidean vector (3, 2): Click to enlarge|250px|right]]   
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If we are given <math>\vec A = 3 \hat{i} + 3 \hat{j} + 2 \hat{k}</math> we know that vector A is 3 units in the x and y direction and 2 units in the z direction. [[Image: VecA.jpg|left|200px]]  
  
  

Revision as of 09:46, 13 June 2011

Vectors are quantities that are specified by both a direction and magnitude of length. The way you draw a vector is by drawing an arrow. This arrow has a length and points in some direction.

Vectora.jpg


When we use vectors we need to introduce the coordinate system. We represent vectors in terms of their components along the coordinate axes. So if we have an arrow we can now communicate with someone else about how much the vector goes in the x direction, the y direction and the z direction by using unit vectors. A vector is a combination of unit vectors that are placed along the the coordinate axes. These unit vector have length one. A unit vector is denoted by a lowercase letter with a hat above it, like so, \hat{a}. It is pronounced as a-hat. Our standard correspond to the x-y-z coordinate system. \hat{i} points along the x-axis \hat{j} points along the y-axis and \hat{k} points along the z-axis. We can express any vector in terms of its components relying on the unit vectors. This helps us understand the direction. As for the magnitude, the length of a vector, we can just measure its length. The magnitude is denoted by \left| \vec A \right|. The magnitude of a vector is a scalar quantity, a numerical value with no direction.
Unitvec.jpg


When we have a vector quantity we put an arrow on top of the labeling letter to remind us that it is a vector. It looks like so \vec A. With \left \langle a_1, a_2, a_3 \right \rangle as any numerical values we can write any vector in the following form \vec A = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}. Another notation for a vector is  \vec A = \left \langle a_1, a_2, a_3 \right \rangle. The components of a vector can be are written as:  \begin{bmatrix} 3 \\ 2\\ \end{bmatrix} the vector has an x-component of 3 and a y-component of 2.


The Euclidean vector (3, 2): Click to enlarge


If we are given \vec A = 3 \hat{i} + 3 \hat{j} + 2 \hat{k} we know that vector A is 3 units in the x and y direction and 2 units in the z direction.

VecA.jpg


If a vector is still a bit abstract to you then think of a compass. The arrow has a certain length, this is our magnitude, and it points in any direction (north, south, east, and west).


Click here for a graphical introduction to vectors:
'Head to tail' addition of two vectors: (3, 2) + (1,-1) = (4, 1): Click to enlarge

When two vectors are added, the sum can be found by placing the two vectors 'head to tail' and finding the coordinate they reach. When a vector is multiplied by a scalar several things may occur:

  • When multiplied by a number with absolute value greater than one, the vector stretches. (Figure 1)
  • When multiplied by a number with absolute value less than one, the vector shrinks. (Figure 2)
  • When multiplied by a negative number, the vector reverses direction. (Figure 3)

Many mathematical and physical entities can be represented by vectors. For example, an object's velocity can be represented by a 'velocity vector', where each component represents the object's speed in a certain direction. Gravity can also be represented by a vector: the strength of gravity's pull corresponds to the magnitude of the vector, and the direction of the pull corresponds to the direction the vector points in. Vector Fields are another important application of vectors.

Figure 1: Stretching a vector A Figure 2: Shrinking a vector A Figure 3: Stretching and reversing the direction of a vector A
Vector bold3.PNG Vector bold4.PNG Vector bold5.PNG
Click here for an algebraic introduction to vectors:

For vectors  \vec{A},\vec{B}, and  \vec{C} and numbers d and e,

The following equalities hold for all vectors and scalars:

  •  \vec{A}+\vec{B} = \vec{B} + \vec{A} (Commutivity of vector addition)
  •  \vec{A}+(\vec{B}+\vec{C}) = (\vec{A}+\vec{B})+\vec{C}  (Associativity of vector addition)
  •  d(\vec{A}+\vec{B}) = d\vec{A} + d\vec{B} (Distributivity of scalars)
  •  d(e\vec{A}) = (de)\vec{A} (Associativity of scalar multiplication)
  • the existence of a zero vector (additive identity),  \vec{0} , such that  \vec{A} + \vec{0} = \vec{A}

The standard algebraic representation of vectors is in terms of components, although not all vectors can be expressed this way:

 \vec{A} = (a_1, a_2, ... , a_k)
 \vec{B} = (b_1, b_2, ... , b_k)

Vectors in this form are added by components:

 \vec{A} + \vec{B} = (a_1, a_2, ... , a_k)+(b_1, b_2, ... , b_k)=(a_1+b_1, a_2+b_2, ... ,a_k + b_k)

Multiplying a vector by a number, known as a scalar, means multiplying each component by that number:

 d\vec{A} = (da_1, da_2, ... , da_k)