Scalars and Vectors - Department of Physics & Astronomy

[Pages:10]Scalars and Vectors

Scalars and Vectors

A scalar is a number which expresses quantity. Scalars may or may not have units associated with them. Examples: mass, volume, energy, money A vector is a quantity which has both magnitude and direction. The magnitude of a vector is a scalar. Examples: Displacement, velocity, acceleration, electric field

Vector Notation

xr ? Vectors are denoted as a symbol with an arrow over the top:

? Vectors can be written as a magnitude and direction:

Er = 15.7 N C@ 30o deg

Vector Representation

? Vectors are represented by an arrow pointing in the direction of the vector. ? The length of the vector represents the magnitude of the vector. ? WARNING!!! The length of the arrow does not necessarily represent a length.

Ar = 2.3m s

Vector Addition

Adding Vectors Graphically.

Ar

Ar

Br

Ar

Br

Br

Cr = Ar + Br

Arrange the

The resultant is drawn

vectors in a head from the tail of the first

to tail fashion.

to the head of the last

vector.

Vector Addition

This works for any number of

vectors.

Ar

Br Cr

Rr = Ar + Br + Cr + Dr

Dr

Vector Addition

Vector Subtraction

Subtracting Vectors Graphically.

Ar

- Br Ar

Br

Br

Flip one vector. Then proceed to add the vectors

Br

( ) Cr = Ar - Br = Ar + - Br

The resultant is drawn from the tail of the first to the head of the last vector.

Vector Components

Any vector can be broken down into components along

the x and y axes.

Example:rr = 5.0m @ 30o from the horizontal. Find its

components.

rr = rrx + rry

rr

rry = r sin^j

rrx = r cosi^

rrx = (5.0m)cos 30oi^

rrx = 4.3mi^

rry = (5.0m)sin 30o ^j

rry = 2.5m^j

Vector Addition by Components

You can add two vectors by adding the components of the vector along each direction. Note that you can only add components which lie along the same direction.

Ar = 3.2 m s i^ + 2.5m s ^j + Br = 1.5m s i^ + 5.2 m s ^j Ar + Br = 4.7 m s i^ + 7.7 m s ^j

Ar + Br = 12.4 m s

Never add the x-component and the y-component

Unit Vectors

Unit vectors have a magnitude of 1. They only give the direction.

y

^j i^ k^ z

A displacement of 5 m in

the x-ddirrec=tio5n mis wi^ritten as

The magnitude is 5m.

The direction is the ?-direction.

x

Finding the Magnitude and Direction

Pythagorean Theorem

r = rx2 + ry2

rr rrx

rry

tan

=

ry rx

=

tan -1

ry rx

Vector Multiplication I: The Dot Product

Ar Br = AB cos The result of a dot product of two vectors is a scalar! Ar

Br

i^ i^ = 1 ^j ^j = 1

i^ ^j = 0 ^j k^ = 0

k^ k^ = 1

i^ k^ = 0

Vector Multiplication I: The Dot Product

( ) ( ) Fr = 2i^ + 3 ^j - 2k^ N sr = 3i^ - 4 ^j - 6k^ m

Fr sr = 2(3)N m + 3(-4)N m + (-2)(-6)N m

Fr sr = 6N m

Vector Multiplication II: The Cross Product

Ar ? Br = AB sin The result of a cross product of two vectors is a new vector!

Ar

Cr

Br

i^ ? i^ = 0 ^j ? ^j = 0

i^ ? ^j = k^ ^j ? k^ = i^

k^ ? k^ = 0 k^ ? i^ = ^j

Vector Multiplication II: The Cross Product

q(vr ? Br)= (qvr ? Br)= (vr ? qBr)

Ar

Cr

Br

Cr = (Br ? Ar)= -(Ar ? Br)

Cr Ar Cr Br

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