Vector & Scalar Quantities

Characteristics of a Scalar Quantity Only has magnitude Requires 2 things:

1. A value

2. Appropriate units

Ex. Mass: 5kg

Temp: 21° C

Speed: 65 mph

Characteristics of a Vector Quantity Has magnitude & direction Requires 3 things:

1. A value

2. Appropriate units

3. A direction!

Ex. Acceleration: 9.8 m/s2 down

Velocity: 25 mph West

More about Vectors A vector is represented on paper by an arrow

1. the length represents magnitude

2. the arrow faces the direction of motion

3. a vector can be “picked up” and moved on

the paper as long as the length and direction

its pointing does not change

Graphical Representation of a VectorThe goal is to draw a mini version of the vectors to give

you an accurate picture of the magnitude and direction. To do so, you must:

1. Pick a scale to represent the vectors. Make it simple yet appropriate.

2. Draw the tip of the vector as an arrow pointing in the appropriate direction.

3. Use a ruler & protractor to draw arrows for accuracy. The angle is always measured from the horizontal or vertical.

Understanding Vector Directions

To accurately draw a given vector, start at the second direction and move the given degrees to the first direction.

N

S

EW

30° N of E

Start on the East origin and turn 30° to the North

Graphical Representation Practice 5.0 m/s East

(suggested scale: 1 cm = 1 m/s)

300 Newtons 60° South of East(suggested scale: 1 cm = 100 N)

0.40 m 25° East of North(suggested scale: 5 cm = 0.1 m)

Graphical Addition of VectorsTip-To-Tail Method

1. Pick appropriate scale, write it down.2. Use a ruler & protractor, draw 1st vector to scale in

appropriate direction, label.3. Start at tip of 1st vector, draw 2nd vector to scale,

label.4. Connect the vectors starting at the tail end of the 1st

and ending with the tip of the last vector. This = sum of the original vectors, its called the resultant vector.

Graphical Addition of Vectors (cont.)

Tip-To-Tail Method

5. Measure the magnitude of R.V. with a ruler. Use your scale and convert this length to its actual amt. and record with units.

6. Measure the direction of R.V. with a protractor and add this value along with the direction after the magnitude.

Graphical Addition of Vectors (cont.)

5 Km

3 Km

Scale: 1 Km = 1 cm

Resultant Vector (red) = 6 cm, therefore its 6 km.

Vector Addition Example #1 Use a graphical representation to solve the

following: A hiker walks 1 km west, then 2 km south, then 3 km west. What is the sum of his distance traveled using a graphical representation?

Vector Addition Example #1 (cont.)

Answer = ????????

Vector Addition Example #2 Use a graphical representation to solve the

following: Another hiker walks 2 km south and 4 km west. What is the sum of her distance traveled using a graphical representation? How does it compare to hiker #1?

Vector Addition Example #2 (cont.)

Answer = ????????

Mathematical Addition of Vectors Vectors in the same direction:

Add the 2 magnitudes, keep the direction the same.

Ex. + =

3m E 1m E 4m E

Mathematical Addition of Vectors Vectors in opposite directions

Subtract the 2 magnitudes, direction is the

same as the greater vector.

Ex.

4m S + 2m N = 2m S

Mathematical Addition of Vectors Vectors that meet at 90°

Resultant vector will be hypotenuse of a

right triangle. Use trig functions and

Pythagorean Theorem.

Mathematical Subtraction of Vectors Subtraction of vectors is actually the addition

of a negative vector. The negative of a vector has the same

magnitude, but in the 180° opposite direction.

Ex. 8.0 N due East = 8.0 N due West

3.0 m/s 20° S of E = 3.0 m/s 20° N of W

Subtraction of Vectors (cont.) Subtraction used when trying to find a change

in a quantity. Equations to remember:

∆d = df – di or ∆v = vf – vi Therefore, you add the second vector to the

opposite of the first vector.

Subtraction of Vectors (cont.) Ex. =Vector #1: 5 km East

Vector #2: 4 km North

4 km N (v2)

5 km W (-v1)

I know it seems silly, but trust me on this one!!!

Component Method of Vector Addition

Treat each vector separately:

1. To find the “X” component, you must:

Ax = Acos Θ

2. To find the “Y” component, you must:

Ay = Asin Θ

3. Repeat steps 2 & 3 for all vectors

Component Method (cont.)

4. Add all the “X” components (Rx)

5. Add all the “Y” components (Ry)

6. The magnitude of the Resultant Vector is

found by using Rx, Ry & the Pythagorean

Theorem:

RV2 = Rx2 + Ry2

7. To find direction: Tan Θ = Ry / Rx

Component Method (cont.)Ex. #1

V1 = 2 m/s 30° N of EV2 = 3 m/s 40° N of W

(this is easy!)

Find: Magnitude & DirectionMagnitude = 2.96 m/sDirection = 78° N of W

Component Method (cont.)Ex. #2

F1 = 37N 54° N of EF2 = 50N 18° N of WF3 = 67 N 4° W of S

(whoa, this is not so easy!)

Find: Magnitude & DirectionMagnitude =37.3 N Direction = 35° S of W