Vectors

= = == = = = = =Overview= Vectors are geometric objects that have both **magnitude** and **direction**. They are used in dimensional space to represent vector quantities, or quantities that have both magnitude and direction, such as velocity, momentum and displacement.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =**Operations**=

Graphically
There are two methods for graphical addition:

For two vectors A and B with vector sum C,


 * Tip To Tail:** Place the origin of vector A at the tip of vector B and then create a vector C with the origin at the origin of A and tip at the tip of B


 * Parallelogram:** Place vectors A and B at the same origin, and then draw lines extending from the tip of A parallel to B, and from the tip of B parallel to A until these lines intersect, and then create a vector C with origin at the common origin of A and B and tip at the intersection point of the two lines drawn.

The sum of vectors A=Ai+Aj+Ak and B=Bi+Bj+Bk can be written as

A+B=(A+B)i+(A+B)j+(A+B)k

The difference of vectors A=Ai+Aj+Ak and B=Bi+Bj+Bk can be written as

A+B=(A-B)i+(A-B)j+(A-B)k

**Dot Product**
The dot product of two vectors A and B can be written as

A•B = IAIIBI cos(x)
Where |A| is the magnitude of A, |B| is the magnitude of B and x is the angle between A and B

This result will always be a scalar quantity. In component form **V=ai+bj+ck,** this operation is performed by multiplying components of vectors in the same direction (i with i, j with j etc.), such that The product of vectors A=Ai+Bj+Ck and B=Di+Ej+Fk can be written as **A • B=(AD)i+(BE)j+(CF)k **

**Cross Product**
The cross product of two vectors A and B can be written as

A×B = IAIIBI sin(x)
Where |A| is the magnitude of A, |B| is the magnitude of B and x is the angle between A and B

This result will always be a vector quantity.

In component form **V=ai+bj+ck,** this operation is performed by multiplying each component of the vectors by each other component in the other vector (i with i, i with j, i with k, j with i, j with j, j with k etc.), and changing the direction of the component according to the right hand rule, and then adding all the components together.

**By A Scalar**
The product of vector A=Ai+Aj+Ak and a scalar B can be written as
 * AB=(AB)i+(AB)j+(AB)k**