INTRODUCTION TO VECTORS - PART 1
Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to firstname.lastname@example.org.
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 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.
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 VectorVector Addition - see #9 in the "Examples" document
A zero vector can be denoted with a boldfaced 0 or. In component form it can be written as v = < 0, 0 >.
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