Vectors and Scalars

Vectors and Scalars I know where I’m going

A scalaris 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 Vectors and Scalars

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

Vectors and Scalars • 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 • 5 m • 30 m/sec, East • 5 miles, North • 20 degrees Celsius • 256 bytes • 4,000 Calories

Vectors and Scalars • 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. Vectors and Scalars

Vectors and Scalars • Vector r • Has a magnitude of 1.5 meters • A direction of  = 250 • E 25 N

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

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

Vectors and Scalars • The Negative of a Vector • Same length (magnitude) opposite direction

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

Vector Addition-Tail to Tip Method • 2 vectors same direction Add the following vectors • d1 = 40 m east • d2 = 30 m east • What is the resultant? d2 = 30 m east d1 = 40 m east d1 + d2 = 70 m east Use scale 1cm = 10m Just add, resultant the sum in the same direction.

2 vectors opposite direction Add the following vectors d1 = 40 m east d2 = 30 m west What is the resultant? Vector Addition-Tail to Tip Method Use scale 1cm = 10m d1 = 40 m east d2 = 30 m west d1 + d2 = 10 m east Subtract, resultant the difference and in the direction of the larger.

Vector Addition-Tail to Tip Method • 2 perpendicular vectors Add the following vectors • v1 = 40 m east • v2 = 30 m north • Find R v2 = 30 m north R • Measure angle with a protractor • Measure length with a ruler • Use scale 1cm = 10m • R = 50 m, E 370 N  v1 = 40 m east

Vector Addition-Tail to Tip Method • 2 random vectors • To add two vectors together, lay the arrows tail to tip. • For example C = A + B Use scale 1cm = 10m link

Vector Addition-Tail to Tip Method • The magnitude of the resultant vector will be greatest when • the original 2 vectors are positioned how? (00 between them) • The magnitude will be smallest when the original 2 vectors • are positioned how? (1800 between them)

Vector Addition-Tail to Tip Method 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. Vector Addition

Vector Addition-Tail to Tip Method link • adding 3 vectors

Vector Addition-Tail to Tip Method • Example: A car travels • 20.0 km due north • 35.0 km in a direction N 60° W • Find the magnitude and direction of the car’s resultant • displacement graphically. Use scale 1cm = 10km We cannot just add 20 and 35 to get resultant vector!!

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

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

Add the following velocity vectors using the parallelogram method V1 = 60 km/hour east V2 = 80 km/hour north Vector Addition-Parallelogram Method

Vector Addition-Parallelogram Method F1 + F2 = F3 = R

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

Vector Subtraction-Tail to Tip Method • Vector Subtraction • Simply add its negative • For example, if D = A - B then use link

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

R B A Vector Addition-Mathematical Method • Adding Vectors Mathematically (Right angles) • Use Pythagorean Theorem • Trig Function Ex: If A=3, and B=5, find R 

Vector Addition-Mathematical Method • I walk 45 m west, then 25 m south. • What is my displacement? A = 45 m west C2=A2 + B2  C2=(45 m)2 + (25 m)2 C=51.5 m C B = 25 m south C = 51 m W 29o S

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

Vector Addition-Mathematical Method Y tan = 15/8  = tan -1 (15/8) = 620 q X

Vector Addition-Mathematical Method

Vectors and Scalars