1. Today we say goodbye to Calculus …
I know this is very sad but remember …
AND
. . .

2. What is our new beginning?
VECTORS

3. Let the adventure begin …

4. So what exactly are VECTORS?

5. What are our learning goals for this course?
Unit 1: Introduction to Vectors
Unit 2: Application of Vectors
Unit 3: Equations of Lines and Planes
Unit 4: The Intersection of Points, Lines and Planes
And if time permits …
MATRICES!!!

6. Unit 1: Introduction to Vectors
Understand what a vector is and how it is represented as a directed line segment.
How to add vectors graphically.
How to multiply a Vector by a Scalar.
Properties of Vectors (Rules of Operations on Vectors)
Vectors in 𝑹 𝟐 & 𝑹 𝟑 (𝒙, 𝒚, 𝐚𝐧𝐝 𝒛)

7. Today’s Learning Goals:
Define a vector and identify whether various quantities are scalars or vectors.
Represent a vector as a directed line segment.
Calculate the magnitude of a vector using the distance formula.
Define equal & opposite vectors and develop techniques to recognize them.

8. What is a vector? WEIGHT TEMPERATURE FORCE MASS VELOCITY HEIGHT SPEED
A scalar is a mathematical quantity having only a magnitude.
A vector is a mathematical quantity having both a magnitude and _________.
direction
Example:
For each of the following, state whether the quantity is a scalar (A) or a vector (B).
WEIGHT
TEMPERATURE
FORCE
MASS
VELOCITY
HEIGHT
SPEED
WEIGHT IS A VECTOR BECAUSE IT IS THE FORCE OF GRAVITY ACTING ON AN OBJECT.
VELOCITY IS A VECTOR
→ IT HAS A MAGNITUDE AND DIRECTION
SPEED IS A SCALAR AS IT DOES NOT HAVE A DIRECTION
TEMPERATURE IS A SCALAR AS IT HAS NO DIRECTION
HEIGHT IS A SCALAR AS IT HAS NO DIRECTION
FORCE IS A VECTOR AS IT HAS A MAGNITUDE AND DIRECTION
MASS IS A SCALAR AS IT DOES NOT HAVE A DIRECTION

9. How do we represent vectors?
To represent vectors we use rays which are directed line segments.
head or tip
tail
head or tip
tail
𝐴𝐵 =200km/h NE
𝐵𝐴 =200km/h SW
The length of the ray is a positive real number which represents the magnitude of the vector.

10. Magnitude
To show magnitude, we place the vector name in absolute value bars.
i.e. 𝐴𝐵 = 𝐵𝐴 =200km/h
Given the points 𝐴(−2, 4) and B(−5,−1), determine the magnitude of 𝐴𝐵 .
Given two points 𝐴( 𝑥 1 , 𝑦 1 ) and B( 𝑥 2 , 𝑦 2 ), the distance between the two points can be calculated using the formula:
(A) 𝐴𝐵 = 74
(A) 𝑑= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1
(B) 𝐴𝐵 =5
(B) 𝑑= 𝑥 1 + 𝑥 − 𝑦 1 + 𝑦 2 2
(C) 𝐴𝐵 = 58
(C) 𝑑= 𝑥 1 + 𝑥 2 2 , 𝑦 1 + 𝑦 2 2
(D) 𝑑= 𝑥 2 − 𝑥 𝑦 2 − 𝑦 1 2
(D) 𝐴𝐵 =7

11. Opposite & Equal Vectors
Two vectors 𝐴𝐵 and 𝐵𝐴 are said to be opposite iff they are parallel and have the same magnitude but opposite directions.
Find all pairs of equal and opposite vectors in the diagram below.
i.e. 𝐴𝐵 = 𝐵𝐴 AND 𝐴𝐵 =− 𝐵𝐴
Two vectors 𝐴𝐵 and 𝐶𝐷 are said to be equal iff they are parallel and have the same magnitude AND the same directions.
i.e. 𝐴𝐵 = 𝐶𝐷 AND 𝐴𝐵 = 𝐶𝐷

12. By the end of today I should be able to …
State the definition of a scalar and vector and determine whether a given quantity is one or the other.
Represent a vector as a ray (a directed line segment) using the proper notation, i.e. 𝐴𝐵
Calculate the magnitude of a vector, 𝐴𝐵 using the distance formula.
State the properties of equal and opposite vectors.
Pick out pairs of equal and opposite vectors from a diagram.
Tomorrow …
VECTOR ADDITION AND SCALAR MULTIPLICATION!
QUESTIONS: p #1, 4, 5, 8, 9, 10abc, 11