6.3 vector in the plane magnitude component form unit vector
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6.3 Vector in the Plane
Magnitude
Component form
Unit Vector
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Vector is a Directed Line Segment
Terminal point
Initial point
Magnitude ( or Length): || PQ ||
Q
P
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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
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Vectors equality
If two vectors equal if they have the same magnitude and direction.
P
Q
R
VRVPQ
3
4PQandRVofSlope
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is a vector in standard position
Vectors in Standard position have an initial point at the origin (0, 0).
PQ
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Component Form of a vector
P = (p1, p2 ); Q = (q1, q2 )
Which can be labeled by just a letter.
2211 , pqpqPQ
2211 , pqpqV
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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
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Find the Component form and Magnitude
Let URS
2,5
4,86)2(4
13)5(8
2
1
u
u
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Find the Component form and Magnitude
Let URS
2,5
4,86)2(4
13)5(8
2
1
u
u
6,13U
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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
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Vector Operations
Scalar Multiplication
Let
21, uKuKUK
30,20
65,45
56,4
KU
UK
KandU
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Vector Operations
Vector Addition
Let
2211 , vuvuVU
2,6
46,24
4,2
6,4
VU
V
U
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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
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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
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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||
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How to Find the Unit Vector
Let
65
7,
65
47,4
65
1
74
7,4
||||
7,4
22
v
v
v
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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
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Direction Angle of a Unit Vector
What is the coordinate of the intersection of the vector and
unit circle?
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Direction Angle of a Unit Vector
What is the slope of the vector?
What function
Is rise over run?
sin,cos
a
b
bjai
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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
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Direction of a Vector can be found if it is not a Unit Vector
jvivvv sincossin,cos
a
b
v
v
cos
sintan
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Homework
Page 436 – 437
# 1, 7, 15, 19,
25, 31, 37, 43,
49, 55, 61, 67, 73
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Homework
Page 436 – 437
# 32, 38, 54, 62,
70, 80