Vectors

Vectors:


The vector represented by the directed line segment (AB) ? has initial pointAand terminal pointB and its lengthis denoted


by|(AB) ? |.

Two vectors are equal if they have the same length and direction.


Component Form

If vis a two-dimensional vector in the plane equal to the vector with initial point at the origin and terminal


point (v_1,v_2), then the component form of v is

v = ?v_1,v_2 ?

The numbers? v?_1 and v_2 are the components of v.

Ifv is a three-dimensional vector equal to the vector with initial point at the origin and terminal point


(v_1,v_2,v_3), then the component form of v is:


v = ?v_1,v_2,v_3 ?


The numbers? v?_1,v_2 and v_3 are the components of v.


Unit Vectors


A vector v of length 1is called a unit vector.


The standard unit vectors are


i = ?1,0,0?,j = ?0,1,0?,k = ?0,0,1?


Any vector v = ?v_1,v_2,v_3 ?can be written as a linear combination of the standard unit vectors as follows:



v = ?v_1,v_2,v_3 ?=v_1 i+v_2 j+v_3 k= (x_2-x_1)i+ (y_2-y_1)j+ (z_2-z_1)k



If v?0, then:


a)v/|v| is a unit vector in the direction ofv (called the direction of the vector v).



b)The equation |v| v/|v| expresses v as its length times its direction.