Vectors

I .  Quiz

II.  Homework

III.   Directed Line Segments and Vectors

A directed line segment is defined as an initial point, P, and a terminal point Q.

Example P=(2,3) and Q=(-1,4)

A vector is the equivalence class of all directed segments of the same length and direction.

We can represent a vector by writing the unique directed line segment that has its initial point at the origin. For example P=(2,3) and Q = (-1,4) is equivalent to the directed line segment "Q-P" = <-3,1>. When we write the <> we mean that the vector has initial point at the origin and terminal point at (-3,1). This notation is called the component form of the vector.

The length of the vector <x,y> is called the norm or magnitude.

We can find it by the formula:

||<x,y>|| = sqrt(x² + y²)

Example:  ||<-3,1>|| = sqrt(1 + 1) = sqrt(10)

We also use the notation -3i + j to denote the vector <-3,1>.

IV. :Unit Vectors in the Direction of v.

A vector is called a unit vector if it has norm = 1. If v = <a,b>, then the unit vector in the direction of v can be found by

1/(||v||) v

Example:  The unit vector in the direction of <-3,1> is

-3/sqrt(10,1/sqrt(10)

We can use the <> notation and the i j notation interchangeably.

V) 3 Dimensional Coordinates

To generalize the plane to 3 dimensions, we draw a third axis, called the z-axis at a right angle from the plane so that if you grab on to the z-axis with your right hand your hand will curl from the positive x-axis to the positive y-axis. (Demonstrate the right hand rule)

To plot a point in the xyz-space We first plot a point in the xy-plane and then draw a segment parallel to the z-axis of length equal to the z coordinate. ( we will Plot (1,2,3) and (2,4,3))

VI) The Distance Formula

The distance between two points (a,b,c) and (d,e,f) and is given by

D = sqrt[(d - a)² + (e - b)² + (f - c)²] 

VII) Vectors

A vector in space is given by <x,y,z> = xi + yj + zk. The algebra rules are similar to those in two dimensions.