INTRODUCTION TO VECTORS - PART 1 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.

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 magnitude and direction and 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 P to point Q. Point P is called the initial point and point Q is called the terminal 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 Notation

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

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

In printed documents, vector names are shown in bold print without the half arrow.

Standard Position of a Vector

A vector with its initial point at the origin of a Rectangular Coordinate System is said to be in standard position. Definition of the Component Form of a Vector

Given a vector in standard position, the coordinates of its terminal point, say , determine its component form, written as where q1 and q2 are called components.

The component form of a vector requires angle brackets .

Finding the Component Form of a Vector - see #1 and 2 in the "Examples" document

If 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 P at and terminal point Q at , its component form is found by .  As a memory aid remember "terminal minus initial" for the x- and y-coordinate!

Magnitude of a Vector - see #3 through 6 in the "Examples" document

This 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 Theorem on a vector in standard position and is

||v|| = .

NOTE: YOU MUST MEMORIZE THIS FORMULA!

Direction of a Vector - see #7 and 8 in the "Examples" document

Place 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 Vector

A zero vector can be denoted with a boldfaced 0 or . In component form it can be written as v = < 0, 0 >.

Vector Addition - see #9 in the "Examples" document

Let u = and v = then u + v = .

The vector u + v is called the resultant vector.

Scalar Multiplication of Vectors - see #10 in the "Examples" document

In vector algebra, any real number is called scalar.

Let u = and let c be a scalar (a real number), then .

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

Given vector v, its negative is v. It has the same magnitude as vector v, but points exactly in the opposite direction.

Vector Subtraction - see #10 in the "Examples" document

Vector subtraction is viewed as an addition of the negative of a vector.

u v = u + ( v) = u 1v