VECTOR ADDITION

Vectors Vectors Quantities have magnitude and direction and can be represented with; 1. Arrows 2. Sign Conventions (1-Dimension) 3. Angles and Definite Directions (N, S, E, W)

Vectors  Every Vector has two parts.  A Head and a Tail  TailHead

Vectors  One dimensional Vectors can use a “+” or “–” sign to show direction m/s   - 45 m  m/s 2  + 60 N 

Picking an Angle for 2-Dimensional Vectors 1. Pick the angle from the tail of the Vector. 2. Find a Definite Direction w.r.t. the Vector (N, S, E, W). 3. Write your angle w.r.t. the Definite Direction.

Addition of Scalars  When adding scalars, direction does not matter so we add or subtract magnitudes.  50 kg + 23 kg = 73 kg  25 s – 13 s = 12 s

Addition of Vectors  Adding vectors is more complicated because the direction affects how the vectors can be combined. A  B A  B  A + B = ?

Addition of 1-Dimensional Vectors One Dimensional Vectors are either in the same or opposite directions.  If the Vectors are in the same direction we add their magnitudes.  If the Vectors are in opposite directions we subtract their magnitudes.

Addition of 2-Dimensional Vectors Addition of 2-Dimensional Vectors requires other methods. 1. Graphical Method 2. Mathematical or Triangle Method 3. Component Method (Best Method)

Properties of Vectors  The addition of vectors yields a RESULTANT (R) R = A + B  Order of addition does not matter! A + B = B + A = R A+B+C = C+B+A = B+C+A = R

Properties of Vectors  Vectors that are Perpendicular to each other are INDEPENDENT of each other.  Ex. Boat traveling perpendicular to the current.  Ex. Projectile thrown in the air.

Graphical Addition of Vectors R = A + B Adding vectors Graphically has certain advantages and disadvantages.  Not a mathematical method

Graphical Addition of Vectors R = A + B Advantages 1. You can add as many vectors as you want Importance: Vector Diagram Disadvantages 1. Need a Ruler and a Protractor 2. Not the most accurate method

Graphical Addition of Vectors R = A + B 1. Pick an appropriate scale to draw vectors 2. Draw the First Vector (A) to scale 3. Draw the Next Vector (B) from the Head of the Previous Vector (A)

Graphical Addition of Vectors R = A + B 4. Draw the Resultant (R) from the Tail of the First to the Head of the Last 5. Measure the Resultant (R), length and direction, and use your scale to give your answer

Vectors are Relative  Vectors that are measured depend on their reference point or frame of reference  Velocities measured are relative w.r.t. the Observer  1-Dimensional (Ex. Bus Ride)  2-Dimensional (Ex. River Crossing)

Subtraction of Vectors R = A – B  Subtraction is the Addition of a Negative Vector A - B = A + (-B)  So what does –B mean?  -B is the vector that has the same magnitude of B but in the opposite direction!

Subtraction of Vectors R = A – B  Ex. A = 50 N  B = 40 N  -B = 40 N   A + B = ?A – B = ?  Does (A – B) = (B – A) ???  No!!! A – B = -B + A = -(B – A)

Vector Addition: Triangle Method Advantages 1. Mathematical Method (More Accurate) Importance: Working with Right Triangles Disadvantages 1. Can only add two vectors at a time

Vector Addition: Triangle Method 1.Use a Vector Diagram to make your triangle (see Graphical Method) 2.See what kind of Triangle you have

Vector Addition: Triangle Method 3.If you have a Right Triangle use:  Pythagorean Theorem  c 2 = a 2 + b 2  Tangent Function  tan  = opposite side = O adjacent side A

Vector Addition: Triangle Method 4. If you do not have a Right Triangle use:  Law of Cosines  c 2 = a 2 + b 2 – 2abCos(C)  Law of Sines  Sin(A) = Sin(B) = Sin(C) a b c

Perpendicular Components of a Vector Every Vector can be split into two perpendicular components  A = A x + A y  A x = Horizontal component of A  A y = Vertical component of A

Perpendicular Components of a Vector A x and A y are one dimensional vectors  A x -  “+”  “-”  A y -  “+”  “-”

To Find Perpendicular Components of a Vector  Since the components(A x, A y ) are perpendicular, they form a right triangle with. Vector A  Use the sine and cosine functions to find A x and A y

To Find Perpendicular Components of a Vector  sin  = Opposite = O Hypotenuse H  cos  = Adjacent = A Hypotenuse H

Component Method Advantages to the Component Method 1. Do not need a ruler or a protractor (More accurate) 2. Can add multiple vectors 3. Every triangle is a right triangle!!

Component Method 1. Break up each vector into its perpendicular components (A x, A y, B x, B y, C x, C y, etc,) 2. Add up all the x-components to get the x-component of Resultant (R x = A x + B x + C x + … etc)

Component Method 3. Add up all the y-components to get the y-component of Resultant (R y = A y + B y + C y + … etc) 4. Recombine R x and R y to make the Resultant, R (R = R x + R y ) (Use the Triangle Method for Right Triangles)  Every triangle is a Right Triangle