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Vectors - Coggle Diagram
Vectors
Introduction to Vectors
Scalar Multiplication of Vectors
Collinear Vectors
Unit Vectors
What are vectors?
Geometric Vectors
Algebraic Vectors
Opposite Vectors
Equal Vectors
Vector Addition
Question
Bob is walking 1.4 m/s due East, and Adam is walking 0.5 m/s N30W.
Draw the following vectors and solve for the resultant.
a)a+b
b)a-b
c)a+2b
Properties of Vectors
Associative Property of Addition
A=(1,2), B=(3,4), C=(2,3)
(A+B)+C=(4,6)+(2,3)=(6,9)
(B+C)+(A)=(5,7)+(1,2)=(6,9)
Distributive Law for scalars
Distributive Property of Addition
A=(1,2), B=(3,4), K=2
2(4,6)=(8,12)
2(1,2)+2(3,4)=(2,4)+(6,8)=(8,12)
Associative Law for Scalars
Commutative Property of Addition
A=(2,7,3), B=(6,5,2)
A+B=(2+6,7+5,2+3)=(8,12,5)
B+A=(6+2,5+7,3+2)=(8,12,5)
Vectors in R2 and R3
In R2
In R3
Operations with Algebraic Vectors in R2
Magnitude of Vectors in R2
Unit Vectors
Operations with Algebraic Vectors in R3
Magnitude of Vectors in R3
Position Vector, OP has it's tail at the Origin and it's head at point p
linear Combination and Spanning Sets
Spanning Sets
If the two vectors A and B are non-collinear vectors, the combination of this set of vectors spans R2 or R3. Any vector in that plane can be written as a linear combination of the original ones
Application of vectors
Vectors as Forces
f=ma
Force itself is a vector because it is the result of multiplying mass (a scalar) with acceleration (a vector)
Velocity
Ground Velocity
The velocity of the waterflow will change the direction of the object.
Air Velocity
The velocity of the airflow will change the direction of the object
The Dot Product
Dot Product of Geometric Vectors
Dot Product of Algebraic Vectors
R2
R3
Properties of the Dot Product
Commutative Property
Distributive Property
Associative Property With a Scalar
Magnitude Property
Scalar and Vector Projections
Scalar Projections
To find magnitude projection
Vector Projections
To find magnitude and direction
Direction Cosines and Direction Angles
Direction Angles: Angles a position vector makes with the x, y, and z axis. We would project the direction vector onto one of the axis
The Cross Product
Properties of Cross Product
Not Commutative
Distributive Law
Scalar Law
AxB=(A3B3-A3B2, A3B1-A1B3, A1B2-A2B1)
Application of Dot and Cross Product
Work
Torque
Area of Parallelogram
Question
A force of 60N is applied at the end of a lever that has a length of 150 cm, if the force is applied at an angle of 27 to the lever, draw a diagram and determine the magnitude of the torque.
In a ballroom, a chandelier is suspended from the ceiling by two wires that make angles of 21 and 45 with the ceiling, if the chandelier weighs 33kg, determine the tension in the wires. (Draw a diagram)
Equation of lines and planes
Vectors and Parametric Equations in R2
Vector Equation of a Line in R2
Parametric Equations of a Line in R2
X and Y values of the Vector Equation
Perpendicular and Parallel Lines
Perpendicular if
m1xm2=0
m1=km2
m1=m2
Parallel if
m1⊥m2
m1*m2=0
Cartesian Equation of a Line
Vector, Parametric, and Symmetric Equations of a Line in R3
Vector Equation
Parametric Equations
Symmetrical Equations
Vector and Parametric Equations of a plane
Vector Equation of a Plane
Parametric Equation
Plane
3 non-collinear points
2 intersecting lines
A line and a point not on the line
2 parallel and non-coincident lines
The Cartesian Equation of a Plane
Question
Given the points A(1,-2,5), B(3,4,-4), and C(5,6,-3), determine the cartesian equation of the plane containing these points, and then find the angle between it and the plane π2: 3x+5y-2z+3=0.
