** 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 physics, such as area, time, and temperature, can be represented by a single number. Other quantities, such as force and velocity, must be represented by two numbers. Specifically, they require both

magnitudeanddirection. To represent such quantities, we use something called a VECTOR.Let's draw a directed line segement (arrow on one end) making a positive angle with a horizontal line. Let's give it a beginning point

and an ending pointPat the tip of its arrow. The length of the directed line segment is the distance betweenQandP.QSince we described the directed line segment in terms of an angle and a distance, we will now call it a VECTOR. Its

is indicated by the positive angle . Itsdirectionis the distance from pointmagnitudeto pointP. From now on we will call PointQthePinitial pointand pointtheQterminal pointof a vector.

NOTE: When drawing a vector, there is always an arrow at the terminal point.

Vector NotationVectors are denoted by lowercase letters, usually u, v, or w, 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 or we could call it simply .

In

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

PQorv.

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 VectorAssume we have a vector in standard position and the coordinates of its terminal point

are .QThese coordinates are then used to write the vector in

component formas , whereandq_{1}are calledq_{2}components.

NOTE: The component form of a vector requires angle brackets .Example:

Write the following vector

vin component form.Using the coordinates of the terminal point, we find .

Finding the Component Form of a Vector- see #1in 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 as follows:Q

NOTE: YOU MUST MEMORIZE THIS FORMULA!For the first component, we subtract from the

x-coordinateof the terminal pointq_{1}theQx-coordinateof the initial pointp_{1 }. Similarly, for the second component, we subtract from thePy-coordinateof the terminal pointq_{2}theQy-coordinateof the initial pointp_{2}.PExample:

Find the component form of a vector with initial point

atPand terminal point(1, 3)atQ

. Name the vector(6, 5)v.Given vector

v= and pointsandP, we can calculateQ

v=<6 (1), 5 (3) >, which equals v = < 5, 8 >.

Magnitude of a Vector- see #2 through 5in the "Examples" documentThe "length" of a vector is called

magnitude. In physics it can 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!Example 3:

Calculate the EXACT magnitude of

v= < 3, 2 >.Given ||

v|| = , we find||v|| =

Direction of a Vector- see #6 and 7in 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 calculate using which we derived in the lesson on polar coordinates.The

x-value is thex-coordinate of the terminal point and they-value is they-coordinate of the terminal point.

Zero VectorA zero vector can be denoted with a boldfaced

0or. In component form it can be written asv=< 0, 0 >.Vector Addition- see #8in the "Examples" document

Let

u= andv= thenu+v= .The vector

u+vis called theresultantvector.

Scalar Multiplication of Vectors- see #9b and 9din 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 #9ain 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 #9cin the "Examples" documentVector subtraction is viewed as an addition of the negative of a vector.

uv=u+ (v)