** INTRODUCTION TO VECTORS - PART 1
Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to
ingrid.stewart@csn.edu.**

Learning Objectives - This is what you must know after studying the lecture and doing the practice problems!

1. Use Vector Notation.

2. Write a vector in component form.

3. Find the magnitude of a vector.

4. Find the direction of a vector.

5. Add vectors.

6. Multiply a vector with a scalar.

7. Find the negative of a vector.

8. Subtract vectors.

Many quantities in geometry and physics, such as area, time, and temperature, can be represented by a single real number. Other quantities, such as force and velocity, involve both

magnitudeanddirectionand cannot be completely characterized by a single real number. To represent such a quantity, we use a directed line segment (arrow) called a VECTOR.Below is the picture of a vector whose direction is indicated by the positive angle , and its magnitude is the distance from point

to pointP. PointQis called thePinitial pointand pointis called theQterminal point.There is always an arrow pointing to the terminal point to indicate that we are looking at a vector and not a line segment.

Vector NotationVectors are denoted by lowercase letters or by their initial and terminal point.

In

handwrittendocuments, a half arrow is placed over either notation. For example, in the picture above we could call the vector .In

printeddocuments, vector names are shown inboldprint without the half arrow.

Standard Position of a VectorA vector

with its initial point at the origin of aRectangular Coordinate Systemis said to be instandard position.

Definition of the Component Form of a VectorGiven a vector in standard position, the coordinates of its terminal point, say , determine its

component form, written as whereandq_{1}are calledq_{2}components.

The component form of a vector requires angle brackets .

Finding the Component Form of a Vector- see #1 and 2in the "Examples" documentIf a vector is not in standard position, we can still find its component form. This also moves it to standard position.

Given a vector with initial point

at and terminal pointPat , its component form is found byQ.

As a memory aid remember "terminal minus initial" for thex- andy-coordinate!

Magnitude of a Vector- see #3 through 6in the "Examples" documentThis is the "length" of the vector. It can also represent speed, weight, etc. The magnitude of a vector

v= is denoted by ||v||.It is found by using the

Pythagorean Theoremon a vector in standard position and is||

v|| = .NOTE: YOU MUST MEMORIZE THIS FORMULA!

Direction of a Vector- see #7 and 8in the "Examples" documentPlace the initial point of a vector at the origin in a coordinate system. Then the direction of the vector is given by the positive angle between the positive

x-axis and the vector.We use the fact that which we derived in the lesson on polar coordinates using the following picture:

Zero VectorA zero vector can be denoted with a boldfaced

0or. In component form it can be written asv=< 0, 0 >.Vector Addition- see #9in the "Examples" document

Let

u= andv= thenu+v= .The vector

u+vis called theresultantvector.

Scalar Multiplication of Vectors- see #10in the "Examples" documentIn vector algebra, any real number is called

scalar.Let

u= and letcbe a scalar (a real number), then .

The Negative of a Vector- see #10in the "Examples" documentGiven vector

v, its negative isv. It has the same magnitude as vectorv, but points exactly in the opposite direction.

Vector Subtraction- see #10in the "Examples" documentVector subtraction is viewed as an addition of the negative of a vector.

uv=u+ (v) =u1v