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A scalar is a quantity described by just a number, usually with units. It can be positive, negative, or zero. Examples: –Distance –Time –Temperature –Speed A vector is a quantity with magnitude and direction. The magnitude of a vector is a nonnegative scalar. Examples: –Displacement –Force –Acceleration –Velocity

Magnitude, what is it?  The magnitude of something is its size.  Not “BIG” but rather “HOW BIG?”

 Magnitude and Direction  When specifying some (not all!) quantities, simply stating its magnitude is not good enough.  For example: ”Where’s the library?,” you need to give a vector! Quantity Category 1.5 m 2.30 m/sec, East 3.5 miles, North 4.20 degrees Celsius bytes 6.4,000 Calories

 Magnitude and Direction  Giving directions: –How do I get to the Virginia Beach Boardwalk from Norfolk? –Go 25 miles. (scalar, almost useless). –Go 25 miles East. (vector, magnitude & direction)

 Graphical Representation of Vectors  Vectors are represented by an arrow.  The length indicates its magnitude.  The direction the arrow point determines its direction.

Vector r  Has a magnitude of 1.5 meters  A direction of  = 25 0  E 25 N

 Identical Vectors  A vector is defined by its magnitude and direction, it doesn’t depend on its location.  Thus, are all identical vectors

 Properties of Vectors  Two vectors are the same if their sizes and their directions are the same, regardless of where they are. x y A B E D C F Which ones are the same vectors? A=B=E=D Why aren’t the others? C: C= - A C: The same magnitude but opposite direction: C= - A F: F: The same direction but different magnitude

 The Negative of a Vector  Same length (magnitude) opposite direction

Tail to Tip Method Steps: i.Draw a coordinate axes ii.Plot the first vector with the tail at the origin iii.Place the tail of the second vector at the tip of the first vector iv.Draw in the resultant (sum) tail at origin, tip at the tip of the 2 nd vector. (Label all vectors) 1.Graphically – Use ruler, protractor, and graph paper. i. Tail to tip method ii. Parallelogram method 2.Mathematically – Use trigonometry and algebra

 2 vectors same direction Add the following vectors d 1 = 40 m east d 2 = 30 m east What is the resultant? d 1 = 40 m east d 2 = 30 m east d 1 + d 2 = 70 m east Just add, resultant the sum in the same direction. Use scale 1cm = 10m

d 1 = 40 m east d 2 = 30 m west d 1 + d 2 = 10 m east Subtract, resultant the difference and in the direction of the larger.  2 vectors opposite direction Add the following vectors d 1 = 40 m east d 2 = 30 m west What is the resultant? Use scale 1cm = 10m

 2 perpendicular vectors Add the following vectors v 1 = 40 m east v 2 = 30 m north Find R v 1 = 40 m east v 2 = 30 m north R  1.Measure angle with a protractor 2.Measure length with a ruler 3.Use scale 1cm = 10m 4.R = 50 m, E 37 0 N

 2 random vectors  To add two vectors together, lay the arrows tail to tip.  For example C = A + B link Use scale 1cm = 10m

1.The magnitude of the resultant vector will be greatest when the original 2 vectors are positioned how? (0 0 between them) 2.The magnitude will be smallest when the original 2 vectors are positioned how? (180 0 between them)

How does changing the order change the resultant? It doesn’t!

 The order in which you add vectors doesn’t effect the resultant.  A + B + C + D + E=  C + B + A + D + E =  D + E + A + B + C  The resultant is the same regardless of the order.

link  adding 3 vectors

We cannot just add 20 and 35 to get resultant vector!! Example : A car travels km due north km in a direction N 60° W 3.Find the magnitude and direction of the car’s resultant displacement graphically. Use scale 1cm = 10km

Parallelogram Method Steps: 1.Draw a coordinate axes 2.Plot the first vector with the tail at the origin 3.Plot the second vector with its the tail also at the origin 4.Complete the parallelogram 5.Draw in the diagonal, this is the resultant. 1.Graphically – Use ruler, protractor, and graph paper. i. Tail to tip method ii. Parallelogram method 2.Mathematically – Use trigonometry and algebra

A B R You obtain the same result using either method. Tail to Tip Method Parallelogram Method

 Add the following velocity vectors using the parallelogram method V 1 = 60 km/hour east V 2 = 80 km/hour north

F 1 + F 2 = F 3 = R

u + v = R Step 1: Draw both vectors with tails at the origin Step 2: Complete the parallelogram Step 3: Draw in the resultant

 Vector Subtraction  Simply add its negative  For example, if D = A - B then use link

 Vector Multiplication  To multiply a vector by a scalar  Multiply the magnitude of the vector by the scalar (number).

 Adding Vectors Mathematically (Right angles)  Use Pythagorean Theorem  Trig Function A B R Ex: If A=3, and B=5, find R 

 I walk 45 m west, then 25 m south.  What is my displacement? A = 45 m west B = 25 m south C C 2 = A 2 + B 2 C 2 = (45 m) 2 + (25 m) 2 C = 51.5 m  C = 51 m W 29 o S

Find the magnitude and direction of the resultant vector below 8 N 15 N R = Resultant Force Vector

X Y  tan  = 15/8  = tan -1 (15/8) = 62 0