Vectors

=**Definitions:**=
 * Vector notation:**

Vectors are usually denoted in **[|lowercase]** boldface, as **a** or lowercase italic boldface, as **//a//**. Other conventions include or //__a__//, especially in handwriting. Alternately, some use a [|tilde] (~) or a wavy underline drawn beneath the symbol, which is a convention for indicating boldface type. If the vector represents a directed [|distance] or [|displacement] from a point //A// to a point //B// (see figure), it can also be denoted as or //__AB__//. []


 * Vector magnitude:**

The magnitude is simply the number of units the vector is (length of the arrow) without the direction. It’s the absolute value of the vector. eg. |**d**| = 10 m. absolute vector = root of (x squared + y squared + z squared) Ex. = (5, 6, 14) absolute vector = Root of (5+6+14) = Root of (25) = 5

//good video explaining how to get magnitude from vector with only the point:// http://www.youtube.com/watch?v=6GoMXuE1FOw

Two vectors are said to be equal if they have the same magnitude and direction. Equivalently they will be equal if their coordinates are equal. So two vectors and are equal if
 * Vector equality:**

[]


 * Relationship between //__AB__// and //__BA__//:**

The magnitude of the two vectors are equal, however the direction is opposite. //__AB__// = //__-BA__//


 * Parallelogram Law of Addition:**

Definition 2: Parallelogram law If two vectors are represented by two adjacent sides of a parallelogram, then the diagonal of parallelogram through the common point represents the sum of the two vectors in both magnitude and direction. Parallelogram law, as a matter of fact, is an alternate statement of triangle law of vector addition. A graphic representation of the parallelogram law and its interpretation in terms of the triangle is shown in the figure :


 * **Parallelogram law** ||
 * [[image:file:///C:%5CDOCUME%7E1%5COwner%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image004.gif width="302" height="263" caption=" Parallelogram law (va2.gif)"]] ||
 * **Figure 6:** ||

[]

The Parallelogram Law
The procedure of "**the parallelogram of vectors addition method**" is
 * draw vector 1 using appropriate scale and in the direction of its action
 * from the tail of vector 1 draw vector 2 using the same scale in the direction of its action
 * complete the parallelogram by using vector 1 and 2 as sides of the parallelogram
 * the resulting vector is represented in both magnitude and direction by the diagonal of the parallelogram

[]

[] []
 * Overall good website for Vecotrs:

__Zero Vector:__**

A zero vector, denoted, is a [|vector] of length 0, and thus has all components equal to zero. It is the [|additive identity] of the [|additive group] of vectors. []

- The sum of two vectors, //Example:// can be written as **C = A + B** []
 * __Addition of Vectors:__**
 * A** and **B**, is a vector **C**, which is obtained by placing the initial point of **B** on the final point of **A**, and then drawing a line from the initial point of **A** to the final point of **B**, as illustrated in Panel 4. This is sometines referred to as the "Tip-to-Tail" method.

- The difference of two vectors, //Example:// or **C = A + (-B)**.Thus vector subtraction can be represented as a vector addition. graphical representation is shown(above), Inspection of this shows that we place the initial point of the vector **-B** on the final point the vector **A**, and then draw a line from the initial point of **A** to the final point of **-B** to give the difference **C**. []
 * __Subtraction of Vectors:__**
 * A - B**, is a vector **C** that is, **C = A - B**

"scalar" =** Any quantity which has a magnitude but no direction associated with it (//Example:// speed, mass and temperature) - The product of a scalar, m say, times a vector **A**, is another vector, **B**, where **B** has the same direction as **A** but the magnitude is changed, that is, //Example://
 * __Scalar Multiplication:__
 * B|** = m**|A|**.

- - refers to the multiplication of a [|vector] by a constant, producing a vector in the same (for ) or opposite (for ) direction but of different length []


 * __Collinear Vectors:

Unit Vector:__** is a [|vector] of length 1, sometimes also called a direction vector (Jeffreys and Jeffreys 1988). The unit vector having the same direction as a given (nonzero) vector  is defined by where  denotes the [|norm] of, is the unit vector in the same direction as the (finite) [|vector]. A unit vector in the direction is given by where  is the [|radius vector]. []
 * [[image:http://mathworld.wolfram.com/images/equations/UnitVector/NumberedEquation1.gif width="44" height="33" caption=" v^^=(v)/(|v|), "]] ||
 * [[image:http://mathworld.wolfram.com/images/equations/UnitVector/NumberedEquation2.gif width="63" height="59" caption=" x_n^^=((partialr)/(partialx_n))/(|(partialr)/(partialx_n)|), "]] ||

-take vector divide by scalar vector = unit vector //Example://
 * //From Class://**

Some properties of vectors, you may remember some of these from common sense, which is good because they are important: (A,B, and C are vectors)

Properties of Vector Addition: Laws of vector addition and scalar multiplication:
 * Communicative Property of Addition: A+B = B+A
 * Associative Property of Addition: A+(B+C) =(A+B)+C
 * Distributive Property of Addition:
 * Additive Identity: 0+A = A
 * Associative Law for Scalars: (AB)C = A(BC)
 * Distributive Law for Scalars: A(B+C) = AB+AC
 * Scalar Multiplicitive Identity: 1 X A = A
 * Additive Inverse: A+(-A) = 0

=**__Vectors as Coordinates__**=


 * Key Terms and Ideas:**

R=Real number

//Two-space R^2// A vector in the Cartesian plane, showing the position of a point //A// with coordinates (2,3).

//Three-space R^3// The vector OP has initial point at the origin O (0, 0, 0)  and terminal point at  P (2, 3, 5). We can draw the **vector** **OP** as follows:

//Unit Vectors (R^2)//


 * i**=//x// **j**=//y//

//Unit Vectors (R^3)//


 * i**=//x// **j**=//y// **k**=//z//

//Right Hand System//

//The difference between a point and vector in two and three-space//

//Representing vectors in two and three-space//

//Adding vectors expressed as cordinates in two and three-space// A + B = (AX + BX)i + (Ay + By)j + (Az + Bz)k

//Multiplying vectors expressed as coordinates by scalar quanitities in two and three-space//

//Linear Combination of vectors in two and three-space//

//Linear independence//

//Spanning Sets//

__Ways of determining position:__

//Cylindrical Coordinates//

//Spherical Coordinates//

//Cartesian...something


 * Dot product:** []


 * Scalar projection:**

The scalar projection of b onto a is the length of the segment AB shown in the figure below.

(I'm sorry about the pictures, they aren't uploading for some reason. This one is the second diagram here: http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/dotprod/dotprod.html)

Thus, mathematically, the scalar projection of b onto a is |b|cos(theta) (where theta is the angle between a and b) which from (*) is given by

(and this one is immediately below the second diagram also here: http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/dotprod/dotprod.html)

This quantity is also called the component of b in the a direction (hence the notation comp).

Basically the same as scalar projection but with a direction.//
 * Vector projection:**

To find the projection of on, draw the line from the "tip" of  perpendicular with. You now have a right triangle with angle between the angles and hypotenuse of length. The length of the projection, the "near side", is then. Since the dot product can be defined as, to get the length of the pojection, we need to get rid of that by dividing by it. The length of the projection of on  is

In order to get the projection vector itself, we need to multiply that length by the unit vector in the direction of, which is, of course,. The vector projection of on  is

//([])

The cross product of two vectors a and b is denoted by a × b.
 * Cross product:**

-only possible in 3space and 7 space.

- formula: AxB=ABsin(theta)n(with a little hat)

where --> is the measure of the smaller angle between a and b (0° --> --> --> 180°), a and b are the magnitudes of vectors a and b, and [Image] is a unit vector perpendicular to the plane containing a and b in the direction given by the right-hand rule as illustrated. If the vectors a and b are parallel (i.e., the angle between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0

The direction of the vector// //n(with a little hat)// //is given by the right-hand rule, where one simply points the forefinger of the right hand in the direction of a and the middle finger in the direction of b. Then, the vector// //n(with a little hat)// //is coming out of the thumb (according to the right hand rule). Using this rule implies that the cross-product is anti-commutative, i.e., b × a = -(a × b). By pointing the forefinger toward b first, and then pointing the middle finger toward a, the thumb will be forced in the opposite direction, reversing the sign of the product vector.

Worksheets from May 18th class // Are the following sets of vectors linearly dependent of independent? a) (2, 3) and (7, -1) b) (2, -3) and (-8, 12) c) (1, 3), (5, -2), and (6, 4) d) (1, 2, 3) and (-2, 4, 3) e) (2, -2, 1) and (-6, 6, -2) f) (1, 2, 4), (3, 0, -1), and (5, -7, 0) g) (5, -2, 4), (-1, 4, 2), and (2, 1, 3)

h) (1, 0, 3), (2, 1, -4), (3, 4, 1), and (2, 1, 1)

The properties of cross product []

The properties of dot product

Lines and Planes









Parametric: [] Examples of 2D: [] Examples in 3D: []