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) 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. 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 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 The terminal end or head

Picture of a Vector Named A Magnitude of A A = 10 Magnitude of A A = 10 Direction of A  = 30 degrees Direction of A  = 30 degrees

The Polar Angle for a Vector Start at the positive x-axis and 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. Start at the positive x-axis and 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. 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 implies B = -A

Multiplication of a Vector by a Number. A 2A2A -3A

Graphical Addition of Vectors ( Head –to -Tail Addition ) To find C = A + B : 1 st Put the tail of B on the head of A. 2 nd 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 : 1 st Put the tail of -B on the head of A. 2 nd 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 B C D A B 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 A x and A y. x y A AXAX AYAY Components A X and A Y can be positive, negative, or zero. The quadrant that vector A lies in dictates the sign of the components. Components are scalars.

When the magnitude of vector A is given and its direction specified then its components can be computed easily x y A AXAX AYAY A X = Acos   A Y = Asin  You must use the polar angle in these formulas.

Example: Find the x and y components of the vector shown if A = 10 and  = 225 degrees. A X = Acos  10 cos(225) = A X = Acos  10 cos(225) = A = (-7.07, -7.07) Note: The components are the coordinates of the point that the vector points to. Note: The components are the coordinates of the point that the vector points to. A y = Asin  10 sin(225) = A y = Asin  10 sin(225) = -7.07

Example: Find A x and A y. A X = Acos  10 cos(30) = 8.66 A X = Acos  10 cos(30) = 8.66 A y = Asin  10 sin(30) = 5.00 A y = Asin  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 (A Y /A X ) + C A =

Example: Find A, and  if A = ( -7.07, -7.07) = = 10  = tan -1 (A Y /A X ) + C = tan -1 (-7.07/-7.07) = 225 degrees  = tan -1 (A Y /A X ) + C = tan -1 (-7.07/-7.07) = 225 degrees

Example: Find A, and  if A = ( 5.00, -4.00) = = 6.40  = tan -1 (A Y /A X ) + C = tan -1 (-4.00/5.00) = 321 degrees  = tan -1 (A Y /A X ) + C = tan -1 (-4.00/5.00) = 321 degrees

A x = Acos  A y = Asin  If you know A and  you can get A x and A y with: If you know A x and A y you can get A and  with: If you know A x and A y you can get A and  with: A vector can be represented by its magnitude and angle, or its x and y components. You can go back and forth from each representation with these formulas: A vector can be represented by its magnitude and angle, or its x and y components. You can go back and forth from each representation with these formulas:

Adding Vectors by Components If R = A + B Then R x = A x + B x and R y = A y + B y 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. Solution: R = A + B = ( 3.00, 2.00) + ( 0, 4.00), so R = ( 3.00, 6.00 ) Solution: R = A + B = ( 3.00, 2.00) + ( 0, 4.00), so R = ( 3.00, 6.00 )

Example If R = A + B find the magnitude and direction of R. 1 st : Find the components of A and B. A x = 10cos 30 = 8.66 A y = 10 sin30 = 5.00 B x = 8cos 135 = B y = 8sin 135 = 5.66 so: A = (8.66,5.00) + B = (-5.66,5.66) ____________ R = ( 3.00, 10.66)

CONNECTION What application of vectors have you seen in real life situations? What application of vectors have you seen in real life situations?

A x = Acos  A y = Asin  If you know A and  you can get A x and A y with: If you know A x and A y you can get A and  with: If you know A x and A y you can get A and  with: Exit: Copy this slide into your notebook If R = A + B R x = A x + B x R y = A y + B y If R = A + B R x = A x + B x R y = A y + B y