Phys211C1V p1

VectorsScalars: a physical quantity described by a single number

Vector: a physical quantity which has a magnitude (size) and direction.

•Examples: velocity, acceleration, force, displacement.

•A vector quantity is indicated by bold face and/or an arrow.

•The magnitude of a vector is the “length” or size (in appropriate units).

The magnitude of a vector is always positive.

•The negative of a vector is a vector of the same magnitude put opposite direction (i.e. antiparallel)

etcoraoror aaa

) of magnitude (the aa

a

Phys211C1V p2

Combining scalars and vectors

scalars and vectors cannot be added or subtracted.

the product of a vector by a scalar is a vector

x = c a x = |c| a (note combination of units)

if c is positive, x is parallel to a

if c is negative, x is antiparallel to a

Phys211C1V p3

Vector addition

most easily visualized in terms of displacements

Let X = A + B + C

graphical addition: place A and B tip to tail; X is drawn from the tail of the first to the tip of the last

A + B = B + A

A

B

X

AB

X

Phys211C1V p4

Vector Addition: Graphical Method of R = A + B•Shift B parallel to itself until its tail is at the head of A, retaining its original length and direction.•Draw R (the resultant) from the tail of A to the head of B.

AB

+ = A

B

=

R

the order of addition of several vectors does not matter

A

CB

D

A

B

C D

D

BA

C

Phys211C1V p5

Vector Subtraction: the negative of a vector points in the opposite direction, but retains its size (magnitude)

• A B = A +( )

AB

= AB

R

= A

B

Phys211C1V p6

Resolving a Vector (2-d)

replacing a vector with two or more (mutually perpendicular) vectors => components

directions of components determined by coordinates or geometry.

A

Ay

Ax

A = Ax + Ay

Ax = x-componentAy = y-component

sincostan

22

AAAAA

A

AAA

yxx

y

yx

AAy

Ax

Be careful in 3rd , 4th quadrants when using inverse trig functions to find .

Component directions do not have to be horizontal-vertical!

Phys211C1V p7

Vector Addition by componentsR = A + B + C

Resolve vectors into components(Ax, Ay etc. )

Add like componentsAx + Bx + Cx = Rx

Ay + By + Cy = Ry

The magnitude and direction of the resultant R can be determined from its components.

in general R A + B + CExample 1-7: Add the three displacements:

72.4 m, 32.0° east of north

57.3 m, 36.0° south of west

72.4 m, straight south

Phys211C1V p8

Unit Vectors

a unit vector is a vector with magnitude equal to 1 (unit-less and hence dimensionless)

in the Cartesian coordinates:

direction in r unit vectoˆdirection in r unit vectoˆdirection in r unit vectoˆ

z

y

x

k

j

i

Right Hand Rule for relative directions: thumb, pointer, middle for i, j, k.

Express any vector in terms of its components:

kjiA ˆˆˆzyx AAA

Phys211C1V p9

Products of vectors (how to multiply a vector by a vector)

Scalar Product (aka the Dot Product)

is the angle between the vectors

A.B = Ax Bx +Ay By +Az Bz = B.A

= B cos A is the portion of B along A times the magnitude of A

= A cos B is the portion of A along B times the magnitude of B

ABBABA

coscosAB

1800

B A

B cos

note: the dot product between perpendicular vectors is zero.

0ˆˆ0ˆˆ0ˆˆ1ˆˆ1ˆˆ1ˆˆ

ikkjji

kkjjii

Phys211C1V p10

Example: Determine the scalar product between

A = (4.00m, 53.0°) and B = (5.00m, 130.0°)

Phys211C1V p11

Products of vectors (how to multiply a vector by a vector)

Vector Product (aka the Cross Product) 3-D always!

is the angle between the vectors

Right hand rule: AB = C

A – thumb

B – pointer

C – middle

Cartesian Unit vectors

sinABC ABBAC

1800

jikikjkji

kkjjiiˆˆˆˆˆˆˆˆˆ0ˆˆ0ˆˆ0ˆˆ

C = AB sin = B sin A is the portion of B perpendicular A times the magnitude of A= A sin B is the portion of A perpendicular B times the magnitude of B

Phys211C1V p12

Write vectors in terms of components to calculate cross product

kji

kkjkik

kjjjij

kijiii

kjikjiBA

ˆ)(ˆ)(ˆ)(

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

)ˆˆˆ()ˆˆˆ(

xyyxzxxzyzzy

zzyzxz

zyyyxy

zxyxxx

zyxzyx

BABABABABABA

BABABA

BABABA

BABABA

BBBAAA

C = AB sin = B sin A is the part of B perpendicular A times A= A sin B is the part of A perpendicular B times B

sinABC ABBAC

B A

B sin

Phys211C1V p13

Example: A is along the x-axis with a magnitude of 6.00 units, B is in the x-y plane, 30° from the x-axis with a magnitude of 4.00 units. Calculate the cross product of the two vectors.