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Starter
If you are in a large field, what two pieces of information are required for you to locate an object in that field?
What is a vector?
A vector is a mathematical quantity with two characteristics:
1. Magnitude or Length
2. Direction ( usually an angle)
Vectors vs. Scalars
A vector has a magnitude and direction.
Examples: velocity, acceleration, force,
torque, etc.
Vectors vs. Scalars
A scalar is just a number.
Examples: mass, volume, time, temperature, etc.
A vector is represented as a ray,or an arrow.
V
The initial end or tail
The terminal end or head
Picture of a Vector Named A
Magnitude of A
A = 10
Direction of A
q = 30 degrees
The Polar Angle for a Vector
Start at the positive x-axisand rotate counter-clockwise until you reach the vector.
That’s how you find the polar angle.
Polar angles are always positive.They go from 0 to 360 degrees.
Two vectors A and B are equal if they have the same magnitude and direction.
A B
This property allows us to move vectors around on our paper/blackboard without changing their properties.
A = -B says that vectors A and B are anti-parallel. They have same size but the opposite direction.
A
B
A = -B also impliesB = -A
Multiplication of a Vector by a Number.
A
2A-3A
Graphical Addition of Vectors( Head –to -Tail Addition )
To find C = A + B :1st Put the tail of B on the head of A.
2nd Draw the sum vector with its tail on the tail of A, and its head on the head of B.
Example: If C = A+B, draw C.
Here’s Vector C
Graphical Addition of Vectors( Head –to - Tail Addition )
To find C = A - B :1st Put the tail of -B on the head of A.
2nd Draw the sum vector with its tail on the tail of A, and its head on the head of -B.
Example: If C = A-B, draw C.
Here’s Vector C = A - B
Addition of Many Vectors
A
BC D
AB
C
D
R
R = A + B + C + D
Add A,B,C, and D
Vector Addition by Components
A vector A in the x-y plane can be represented by its perpendicular components called Ax and Ay.
x
y
A
AX
AY
Components AX and AY
can be positive, negative,or zero. The quadrantthat vector A lies indictates the sign of thecomponents.Components are scalars.
When the magnitude of vector A is given and its direction
specified then its componentscan be computed easily
x
y
A
AX
AYAX = Acosq
AY = Asinq
You must use the polar angle in these formulas.
Example: Find the x and y components of the vector shown ifA = 10 and q = 225 degrees.
AX = Acos = q 10 cos(225) = -7.07
A = (-7.07, -7.07)
Note: The components are the coordinates of the point that the vector points to.
Ay = Asin = q 10 sin(225) = -7.07
Example: Find Ax and Ay.
AX = Acos = q 10 cos(30) = 8.66Ay = Asin = q 10 sin(30) = 5.00
A = (8.66, 5.00)
The magnitude and polar angle vector can be found by knowing its components
= tan-1(AY/AX) + C
A =
Example: Find A, and q if A = ( -7.07, -7.07)
== 10
= tan-1(AY/AX) + C = tan-1(-7.07/-7.07) + 180 = 225 degrees
Example: Find A, and q if A = ( 5.00, -4.00)
== 6.40
= tan-1(AY/AX) + C = tan-1(-4.00/5.00) + 360 = 321 degrees
Ax = Acosq
Ay = Asinq
If you know A and , q you can get Ax and Ay with:
If you know Ax and Ay
you can get A and q with:
A vector can be represented by its magnitude and angle, or its x and y components. You can go back and forthfrom each representation with these formulas:
Adding Vectors by Components
If R = A + B
Then Rx = Ax + Bx
and Ry = Ay + By
So to add vectors, find their components and add the like components.
Example
A = ( 3.00, 2.00 ) and B = ( 0, 4.00)
If R = A + B find the magnitude and direction of R.
Then R = = 6.70
q = tan-1( 6/3) = 63.4o
Solution: R = A + B = ( 3.00, 2.00) + ( 0, 4.00),
so R = ( 3.00, 6.00 )
ExampleIf R = A + B find the magnitude and direction of R.
1st: Find the components of A and B.
Ax = 10cos 30 = 8.66 Ay = 10 sin30 = 5.00 Bx = 8cos 135 = -5.66 By = 8sin 135 = 5.66
so: A = (8.66,5.00) + B = (-5.66,5.66) ____________ R = ( 3.00, 10.66)
3rd: Get R and q : R = = 11.1 q = tan-1 ( 10.7/3.00) = 74.3o
CONNECTION
What application of vectors have you seen in real life situations?
Ax = Acosq
Ay = Asinq
If you know A and , q you can get Ax and Ay with:
If you know Ax and Ay
you can get A and q with:
Exit: Copy this slide into your notebook
If R = A + B Rx = Ax + Bx
Ry = Ay + By