Vectors

6001

INTRODUCTION

SCALAR QUANTITIES: _______________________________________________________

VECTOR QUANTITIES: ________________________________________________________

Geometry:

Line Segment Vector

Magnitude only

Magnitude and direction

length

𝑑=√ (𝑥2−𝑥1 )2+( 𝑦2− 𝑦1 )2 h𝑙𝑒𝑛𝑔𝑡 =𝑑√ (𝑥2− 𝑥1 )2+ (𝑦2 − 𝑦1 )2

𝜃=arctan( 𝑦𝑥 )

x is negative add

y only negative add

Vocabulary and Notation:

Standard Position:

Magnitude (____________)

Amplitude /Direction Angle (____________)

Bearing:

Equal Vectors :

Opposite Vectors:

Zero Vector:

Unit Vector:

�⃑�𝐵

Initial point and terminal point (where arrow is)

Initial point origin

length

Counter clockwiseFrom the x-axis

Clockwise from the North

Same direction, same magnitude

Same magnitude, opposite direction

A

B

B

A

0 magnitude called a point vector

Length 1

Component Form:

Horizontal = x

Vertical = y

,x y

,Horizontal Vertical

, chevrons

Component Form: Graphically

x

y

count

⟨ 6,3 ⟩

Component Form: Numerically

Vector has endpoints A ( 4, -6) and B ( -6 , 1 ).

Find the component form of

v

v

Vector has component form and initial point A ( -3, -2)

Find the terminal point .

5,3v

v

⟨ −6 − 4,1 −(−6 )⟩⟨ −10,7 ⟩

B(x,y)

𝑥− (− 3 )=−5

𝑥+3=−5𝑥=− 8

𝑦− (− 2 )=3𝑦+2=3𝑦=1

𝐵 (−8,1 )

Examples: A

Sketch the vector A with Magnitude : 50 mmAmplitude: θ = 20o

Sketch the vector B with Magnitude : 70 mmBearing: θ = 135o

Examples: B

Protractor Skills:

Measure the magnitude and amplitude.

Measure the magnitude and amplitude.

A

DOT Paper

Sketch the vector C with Initial point ( 0 , 0 ) and Terminal point ( -3 , 5 )

Find the component form of the vector.

Sketch the vector D with Initial point ( 0 , 0 )

and Terminal point ( 6 , -2 )

Find the component form of the vector.

Examples:

Sketch the vector E with Initial point ( 4 , 5 ) and Terminal point ( -1 , -3 )

Find the component form of the vector.

Sketch the vector F with Initial point ( -2 , -3 ) and Terminal point ( 4 , 1 )

Find the component form of the vector.

⟨ −1− 4 , −3 −5 ⟩⟨ −5 , −8 ⟩

⟨ 4− (− 2 ) ,1 −(−3) ⟩⟨ 6,4 ⟩

x

y

Find the component form of the vector.

Draw the vector in standard position.

Standard Position:

count ⟨ 5 , −4 ⟩

numerically

⟨ 1−(− 4 ) ,1 −5 ⟩⟨ 5 , −4 ⟩

Component Form:

x

y,x y

and v

cos( ) x cos( )

sin( ) sin( )

xv

v

yy v

v

TRIG

𝜃

𝑣

magnitude

Find the component from of a vector with

41 60ov

16 140ov

Trig component form

⟨ 41 cos 60 ° , 41sin 60 ° ⟩Standard component ⟨ 41

2,41√3

2 ⟩⟨ 16 cos140 ° , 16 sin 140 ° ⟩trig

standard

Vectors

6002

Magnitude and Amplitude:

Magnitude is the length of the vector.v

2 2v x y

Amplitude θ is the direction angle - Rem: positive direction

arctany

x

Rem: if x is negative add 180o

if y-only is negative add 360o

Find the magnitude and amplitude of the vector:

7, 6

if ( 3,2) and (2, 10)

v

AB A B

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Unit Vector in the direction of :

1u

To find the unit vector in the direction of

Divide the vector by its magnitude.

v

2 23,4 3 4 5

3 4,

5 5

v v

u

Adding and Subtracting Vectors

Protractor Skills:

Adding: Heel to Toe

a

b

a+b

⟨ −8,3 ⟩

⟨ 3,5 ⟩⟨ −5,8 ⟩

DOT Paper Adding: Heel to Toe

4,7

8,4

a

b

Find a b

a

b

⟨ − 4,11⟩

Adding and Subtracting Vectors

Protractor Skills:

Subtracting: Toe to Toe - must point toward the subtracting vector

a

b

a - b b - a

⟨ 4,4 ⟩

⟨ − 4,7 ⟩

⟨ 8 , −3 ⟩

⟨ −8,3 ⟩

5,4

3,8

a

b

DOT Paper Subtracting: Toe to Toe

Find a b

a

b