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6.3 Vector in the Plane
Magnitude
Component form
Unit Vector
Vector is a Directed Line Segment
Terminal point
Initial point
Magnitude ( or Length): || PQ ||
Q
P
Let P = (0,0) and Q = (3,4)
To find the Magnitude || PQ ||
Direction (slope) is always important.Slope of
525169
0403 22
PQ
PQ
3
4
03
04
PQ
Vectors equality
If two vectors equal if they have the same magnitude and direction.
P
Q
R
VRVPQ
3
4PQandRVofSlope
is a vector in standard position
Vectors in Standard position have an initial point at the origin (0, 0).
PQ
Component Form of a vector
P = (p1, p2 ); Q = (q1, q2 )
Which can be labeled by just a letter.
2211 , pqpqPQ
2211 , pqpqV
Vector “V” can renamed
If || V || = 1, then V is a Unit Vector. || V || = 0 iff V is 0
2211 , pqpqV
222
1
21,
vvV
vvV
Find the Component form and Magnitude
Let URS
2,5
4,86)2(4
13)5(8
2
1
u
u
Find the Component form and Magnitude
Let URS
2,5
4,86)2(4
13)5(8
2
1
u
u
6,13U
Find the Component form and Magnitude
Let URS
2,5
4,86)2(4
13)5(8
2
1
u
u
6,13U
3.14205
613 22
U
U
Vector Operations
Scalar Multiplication
Let
21, uKuKUK
30,20
65,45
56,4
KU
UK
KandU
Vector Operations
Vector Addition
Let
2211 , vuvuVU
2,6
46,24
4,2
6,4
VU
V
U
Parallelogram Law used in Addition of VectorsGraph the
Vectors
move the tail
of one vector to
the head of
the other
vector.
6,4
4,2
Parallelogram Law used in Addition of VectorsGraph the
Vectors
move the tail
of one vector
to the head
of the other
vector.
6,4
4,2
2,6
Properties of Vectors
U + V = V + U (Comm.)(U + V) + W = U + (V + W) (Asso.)U + 0 = U (Identity)U + (-U) = 0(Inverse)C(DU)=(CD)U (Comm.)(C + D)U = CU + DU (Dist.)1(U)=U; 0(U)=0|| cV|| =|c| x ||V||
How to Find the Unit Vector
Let
65
7,
65
47,4
65
1
74
7,4
||||
7,4
22
v
v
v
Standard Unit Vector
Writing the Unit Vector as Standard Unit Vector.
i =
j =
0,1
1,0
65
7,
65
4
||||
7,4
v
vu
v
0,0 1
1
i
j
jiu65
7
65
4
Direction Angle of a Unit Vector
What is the coordinate of the intersection of the vector and
unit circle?
Direction Angle of a Unit Vector
What is the slope of the vector?
What function
Is rise over run?
sin,cos
a
b
bjai
Direction Angle of a Unit Vector
What is the slope of the vector?
What function
Is rise over run?
sin,cos
a
b
bjai
a
b
cos
sintan
ji sincos
Direction of a Vector can be found if it is not a Unit Vector
jvivvv sincossin,cos
a
b
v
v
cos
sintan
Homework
Page 436 – 437
# 1, 7, 15, 19,
25, 31, 37, 43,
49, 55, 61, 67, 73
Homework
Page 436 – 437
# 32, 38, 54, 62,
70, 80
