VectorVector

A quantity that shows both A quantity that shows both magnitude and direction.magnitude and direction.

Vector ExamplesVector Examples

PositionPositionDisplacementDisplacementVelocityVelocityAccelerationAccelerationForceForceMomentumMomentum

Vectors (One dimension)Vectors (One dimension)

vector = +/- scalarvector = +/- scalar

direction magnitude

• +: right, up, north, east -: left, down, south, west

ex: +3.9 m/s, -5.2 N, -76 m

Vectors (Two Dimensional)Vectors (Two Dimensional) v = vector v = magnitude, angle direction v = vector v = magnitude, angle direction

= magnitude @ angle direction = magnitude @ angle direction = magnitude = magnitude (called magnitude-angle form)(called magnitude-angle form) A, v, dA, v, d11, F, FRR In a book a vector is represent as a bold letter, e.g. In a book a vector is represent as a bold letter, e.g. AA

Examples:Examples:v=3.4 m/s, 25v=3.4 m/s, 25º = º = 3.4 m/s @ 253.4 m/s @ 25º = 3.4 m/sº = 3.4 m/s

F=8.2 N, -64º = 8.2 N @ -64º = 8.2 N F=8.2 N, -64º = 8.2 N @ -64º = 8.2 N

d=47 m, 15º N of W = 47 m @ 15º N of W = 47 m d=47 m, 15º N of W = 47 m @ 15º N of W = 47 m

angle direction

25°

-64°

15° N of W

Graphically Representing a Vector Graphically Representing a Vector

θ

A

A=A, θ

A=magnitude of vector A = length of vector A

The tail of the vector

The head (tip) of the vector

A vector can be moved; as longas the magnitude and directionare the same the vector is unchanged.

A

Negative of a Vector Negative of a Vector (Opposite Direction of a Vector)(Opposite Direction of a Vector)

θ+180°

-A

A=A, θ

A = 5.6 m/s, 60° -A= 5.6 m/s, 240°

-A=A, θ±180

The negative of a vector has the same magnitude as the original vectorwith a 180° difference in direction

A

Same magnitude, but a direction difference of 180°

Scaling a VectorScaling a Vector

A=A,A=A,θθ aA=aA, aA=aA, θθ (To scale a vector, only multiply the (To scale a vector, only multiply the

magnitude of the vector by the factor; the magnitude of the vector by the factor; the angle is unchanged).angle is unchanged).

A=4.0 m/s, 23°A=4.0 m/s, 23°

2A=8.0 m/s, 23°2A=8.0 m/s, 23° Twice as long as the original vector,but in the same direction.

Vector Direction ConventionsVector Direction Conventions

0º

90º

180º

270º

-270º

-180º

-90º

E

S

N

WE

N

W

S

E of N

N of E

S of E

E of SW of S

S of W

N of W

W of N

Example of Vector Direction Example of Vector Direction ConventionsConventions

A

A=3.4 km, 57°

B

B = 7.9 m/s, 18° S of W

Component Form of a VectorComponent Form of a Vectory)(v x)(v v yx

y)(vy

x)(v x

v

vx

vy

v

direction)- xin thedirection a magnitude a gives(it r unit vecto- x x

direction)-y in thedirection a magnitude a gives(it r unit vecto-y y

Example of the Component Form of a VectorExample of the Component Form of a Vector

y)m/s (3.0 x)m/s (4.0 v

y)m/s (3.0

x)m/s (4.0

v

4.0 m/s

3.0 m/s

v

direction)- xin thedirection a magnitude a gives(it r unit vecto- x x

direction)-y in thedirection a magnitude a gives(it r unit vecto-y y

Conversion of a Vector in Magnitude-Angle Conversion of a Vector in Magnitude-Angle Form to Component FormForm to Component Form

v, v

37 m/s, 5.0 v

y)sin (v x) cos (v v

y)m/s (3.0 x)m/s (4.0 v

y)37sin m/s (5.0 x) 37 cos m/s (5.0 v

y)m/s (3.0 x)m/s (4.0 37 m/s, 5.0 v

Conversion of Component Form to Conversion of Component Form to Magnitude-Angle Form of a VectorMagnitude-Angle Form of a Vector

m/s 9.4 m/s) 0.8((-5.0m/s) v 22

y)(v x)(v v yx

x

y1

v

vtan

v

magnitude of v )v()(v v 22x y

direction of v

adjust angle for quadrant, If necessary

y)m/s (-8.0 x)m/s (-5.0 v Example:

quadrant) (3rd 238 58 m/s 5.0-

m/s 8.0-tan 1

238 m/s, 9.4 y)m/s (-8.0 x)m/s (-5.0 v

238 m/s, 4.9

x)(v x

y)(vy