Scalar & Vector Quantities

SCALAR QUANTITIES

•Described by a single number and unit of measurement.•Gives the magnitude (size)Examples

Mass = 20 gTime = 20.0 sTemperature = 20oC Speed = 20 m/s

VECTOR QUANTITIES••Arrows are usedArrows are used••Described by a single number and aDescribed by a single number and a unit of measurement (scalar)unit of measurement (scalar)••Indicated direction. (head of arrow)Indicated direction. (head of arrow)

ExamplesExamples30 m/s, East30 m/s, East30 m/s, N of E30 m/s, N of E

DETERMINING DIRECTION

B A

CD

N of E

N of W

S of WS of E

ADDITION OF COLLINEAR VECTORS

Resultant vector Resultant vector represent the total of two or more represent the total of two or more

vectorsvectorsdrawn from the tail of the lst vectordrawn from the tail of the lst vector to the head of the last vectorto the head of the last vector

VECTORS ACTING IN THE SAME DIRECTION

Connect them head-to-tail and add

Always indicate the direction of the the resultant

10 km 10 km

R = 20 km, 0o E

VECTORS ACTING IN OPPOSITE DIRECTIONS

Connect them tail to tail Connect them tail to tail

SubtractSubtract

20 km 10 km

R = 10 km, 0o, W

VECTORS ACTING AT ARIGHT ANGLE TO EACH

OTHER

••Connect head-to-tailConnect head-to-tail••Draw resultant from the tail of the lst Draw resultant from the tail of the lst vector to the head of the second vector.vector to the head of the second vector.••Determine resultant using pythagorean Determine resultant using pythagorean theoremtheorem••Determine angular direction by using theDetermine angular direction by using the tan functiontan function

Pythagorean Theorem

• C2 = a2 + b2

• C = hypotenuse

• A & B are sides

20 km

10 km

or20 km

10 km

10 km

20 km

R2 = (10 km)2 + (20 km)2

R = 22 km = 20 km

Determine angle makes with the X-axis.Use the correct number of significant figures.

Tan = Y X

= 20 km 10km

= 2

= 63o = 60o

Trig of the Right Triangle

Ө

hypotenuse

Adjacent side

Opposite side

SOH

Sin Ө = opposite side

hypotenuse

CAH

Cos Ө = adjacent side

hypotenuse

TOA

• Tan Ө = opposite side

adjacent side

NON-COLLINEAR VECTORS

A.A. Draw vectors on x-y axisDraw vectors on x-y axisB.B. Determine the X & Y components of Determine the X & Y components of each vectoreach vector

a. X-component = use Cos functiona. X-component = use Cos function b. Y-component = use Sin functionb. Y-component = use Sin function

c. Theta is 0c. Theta is 0oo if vector is located on X-axis if vector is located on X-axis d. Theta is 90d. Theta is 90oo if vector is located on Y-axis if vector is located on Y-axis

C. Indicate direction of vectors (+) and (-)C. Indicate direction of vectors (+) and (-)

D. Add X-components

E. Add Y-components

F. Draw vectors representing the sum of the components head-to-tail. Signs (+) and (-) represents the direction of vectors.

G. Draw Resultant

H. Calculate R using Pythagorean Theorem

I. Calculate theta

J. Indicate direction relative to N, E, S or W

EXAMPLE

Walk due west for 52 paces, then walk 30.0Walk due west for 52 paces, then walk 30.0oo North of West for 42 paces, and then walkNorth of West for 42 paces, and then walk due north for 25 paces.What are thedue north for 25 paces.What are the magnitude and direction of the magnitude and direction of the resultant, Rresultant, R

VECTOR DIAGRAM

42 25

52

TABLEPACESPACES ØØ XX YY

5252 0.00.0 -52-52 00

4242 30.030.0 -36-36 2121

2525 90.090.0 00 2525

SumSum -88-88 4646

DETERMINING DIRECTION OF RESULTANT

DRAW TRIANGLE USING SUMS OF X- DRAW TRIANGLE USING SUMS OF X- AND Y-COMPONENTSAND Y-COMPONENTS

R

88 Paces

46 Paces N of W

VALUE OF RESULTANT

Use Pythagorean TheoremUse Pythagorean Theorem

RR22 = (88 paces) = (88 paces)22 + (46 paces) + (46 paces)22

R = 99 pacesR = 99 paces

Angular Direction

Use tan functionUse tan function

tan tan = 46 paces = 46 paces 88 paces88 paces

Ø = 28Ø = 28oo

FINAL ANSWER

99 Paces, 2899 Paces, 28oo, N of W, N of W

Units of Measurements

MKS: meters (m), Kilogram (kg), second (s)

kilometer (km), hour (h)

CGS: centimeter (cm), gram (g), second (s)

FPS: foot (ft.), pound (lb.), second (s)

miles (mi.), hour (h)