** INTRODUCTION TO VECTORS - PART 2
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. Find the dot product of two vectors.

2. Find unit vectors.

3. Express vectors in terms of the i and j unit vectors.

4. Find the angle between two vectors.

5. Determine if two vectors are parallel, perpendicular, or equivalent.

Let's continue our introduction to vectors!

Dot Product- see #1in the "Examples" documentGiven two vectors

u= andv= , theirDot Productis defined to be=

is pronounced

u dot v! TheDot Productis a scalar and not a vector!

NOTE: YOU MUST MEMORIZE THIS FORMULA!

Unit Vector- see #2 and 3in the "Examples" documentA vector with magnitude

1. Standard unit vectors with their special namesis called a1unit vector.Following are some frequently used unit vectors.iandj:

i= and andj=

NOTE: YOU MUST MEMORIZE THESE VECTORS!In handwritten documents you will often see and . There is a little "hat" over the

iandjinstead of a half arrow. A lot of times you will hear this pronounced as "i hat" and "j hat."2. Unit vectors with the same direction as a given nonzero vector

v= .We find these vectors by multiplying a vector

vby a scalar comprised of the magnitude ofvas follows;

NOTE: YOU MUST MEMORIZE THIS FORMULA!

Expressing a Vectorin Terms of the Standard Unit Vectors- see #4in the "Examples" documentIn many applications it is necessary to express vectors in component form in terms of standard unit vectors. This is done as follows:

v= =i+jThe scalars and are called the

horizontalandvertical componentsof vectorv, respectively.

Angle Between Two Vectors- see #5in the "Examples" documentThe

Law of Cosinesis used to develop a formula to find the angle between two vectors. The angle is always betweenand0^{o}^{}.180^{o}^{}Given a vector

uand a vectorv, the angle is found via the cosine ratio and its inverse.That is,

The numerator in this fraction is a

Dot Product, but the denominator is a simple multiplication of two real numbers!

Orthogonal, Parallel, and Equivalent Vectors- see #6in the "Examples" documentTwo vectors are orthogonal (

between them) if their90^{o}DotProductequals0.Two nonzero vectors

uandvare parallel if there is some scalarc, such thatu=cv.Two vectors that have the same magnitude and direction are equivalent.