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:
�⃑�𝐵Initial 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 )=−5𝑥+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( )
x vv
y y vv
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
arctan yx
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
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 VectorsProtractor 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 VectorsProtractor 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