Vectors
Introduction to Vectors
Applications of Vectors
Vector vs Scalar
Vectors
Scalar
Properties of Vectors 
Multiplication of a vector by a Scalar (k) 
Geometric Vectors: Vectors that are considered without reference to coordinate axes
Algebraic Vectors: Vectors that are placed on a coordinate plane in order to use vectors in application
Vectors as Forces:
Equations of Lines & Planes
In R^2 Equation of Lines 
Unit Vectors 
Colinear Vectors 
Any quantities described by the magnitude and direction
Examples: Velocity, Friction, Gravity, Acceleration
Any quantities described by only the magnitude
Examples: Age, Height, Area, Temperature
Drawing & Notation
Notation 
Opposite Vectors: 
Equal Vectors: 
Example: 
Important Distinctions: 
Lie along the positive x,y and z-axes and they are used to write positive vectors in 2 ways
Vector Subtraction
Vector Addition
Parralellogram Law
Linear Combination 
Spanning Sets 
Adding and Subtracting vectors uses the same process in both R^2 and R^3, of course in R^2 you ignore the z-component
CH 6: 
Worked Example #3
Vectors in R^3
Vectors in R^2
xy-Plane 
Position Vector 
Magnitude + Component 
xyz-Plane 
Position Vector 
Magnitude + Component 
Example 
CH 6: Example #1 
CH 6: Worked Example #2 
Resolution of a Vector 
Equilibrant Forces:
A number of vector forces that oppose when acting on an object. This force maintains the object in a state of equilibrium
CH 7: Worked Example #2 : 
Velocity
Air Velocity: As an object travels, the velocity of air flow will create resultant velocity and/or change the direction of
the object.
Ground Velocity: As an object travels, the velocity of water flow will create resultant velocity and/or change the
direction of the object.
Scalar & Vector Projection 
Direction Cosines and Direction Angles 
Scalar: 
Vector 
Dot Product
Definition: Scalar number that represents the amount one vector travels in the direction of the other vector. Can be
determine with either geometric or algebraic vectors. Both will provide the same information.
Dot Product of Geometric Vectors
Dot Product of Algebraic Vectors
Definition 
Signs of Dot Product 
Angles 
Sign of Dot Product 
Cross Product
Definition: Given 2 vectors, it is possible to find a third vector, in R^3, that is perpendicular/orthagonal to both, by easily finding the cross product of the 2 vectors
Right-Handed System 
Properties 
Torque 
Area of Parallelogram & Triangle 
CH 7: Worked Exampe #1 
Perpendicular 
>The Dot Product gives a scalar
Parallel
When 2 vectors are Crossed, it gives another vector
CH8: 
Worked Example #2
Recall: In order to find an equation of a line y=mx + b, it requires 2 pieces of information on:
1) Slope/Direction
2) a Point on the line
Parametric 
Vector 
Cartesian 
Angles 
In R^3 
Vector
Plane 
Line 
Parametric
Line 
Plane 
Symmetric
Line 
Plane 
- A line and a point not on the line
- 3 Non-collinear Points
- 2 Intersecting Lines
- 2 Parallel and non-coincident lines
Cartesian Equation of Plane 
CH 8: Worked Example #1 
CH 8: 
Worked Example #3
Zero Vector: A vector with zero magnitude and zero direction
Angles: 
Example: 
Definition: -Force is determined by multiplying mass(kg) and acceleration (m/s^2).
-The resulting unit is in Newtons (N) Since force is calculated by multiplying a scalar (mass) and vector(acceleration), force itself is a vector
Vectors have a Horizontal(i) and Verticle (j) component 
ONLY defined for vectors in R^3 as it is not possible to find a vector perpendicular to 2 non-collinear vector in R^2
Properties 
In R^3 & R^2 
Work 
