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Fun with Vectors. Definition. A vector is a quantity that has both magnitude and direction Examples?. v, v , or AB. Represented by an arrow. B (terminal point). A (initial point). If two vectors, u and v , have the same length and direction, we say they are equivalent. v. u. - PowerPoint PPT Presentation
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Fun with Vectors
Definition
• A vector is a quantity that has both magnitude and direction
• Examples?
Represented by an arrow
A (initial point)
B (terminal point)
v, v, or AB
If two vectors, u and v, have the same length and direction, we say they are equivalent
uv
ab
Vector addition
ab
Vector addition: a + b
a
b
Vector addition: a + b
a+b
a
b
Vector addition: a + b
a+b
Scalar Multiplication
a
Scalar Multiplication
a2a
-a -½ a
Subtraction
ba
Subtraction
ba
-b
Subtraction
ba
-ba+(-b)
Subtraction
ba
-ba+(-b)
Subtraction
ba
-b
a+(-b)
If a and b share the same initial point, the vector a-b is the vectorfrom the terminal point of b to the terminal point of a
Let’s put these on a coordinate system
We can describe a vector by putting its initial point at the origin.
We denote this as a=<a1,a2>
where (a1,a2) represent the terminal point
Graphically
x
y
a=<a1,a2>
(a1,a2)
y
x
z
v=<a,b,c>
a
b
c
(a,b,c)
Given two points A=(x1,y1) and B=(x2,y2),
The vector v = AB is given by
v = <x2 - x1, y2 - y1>
…or in 3-space, v = <x2 - x1, y2 - y1, z2 - z1>
Graphically
A=(-1,2)B=(2,3)
A
B
v = <2-(-1), 3-2> = <3,1>
v
Recall, a vector has direction and length
Definition: The magnitude of a vector v = <x,y,z> is given by
222 zyxv
Properties of VectorsSuppose a, band c are vectors, c and d are scalars
1. a+b=b+a2. a+(b+c)=(a+b)+c3. a+0=a4. a+(-a)=05. c(a+b)=ca+cb6. (c+d)a=ca+da7. (cd)a=c(da)8. 1a=a
Standard Basis Vectors
1
1,0,0
0,1,0
0,0,1
kji
k
j
i
Definition: vectors with length 1 are called unit vectors
Example: We can express vectors in terms of this basis
a = <2,-4,6> a = 2i -4j+6kQ. How do we find a unit vector in the same direction as a?
A. Scale a by its magnitude
11
1
aa
aa
v
aa
v
6,4,2561
6,4,26)4(2
1
1
222
aa
v
Examplea = <2,-4,6>
12.3 The Dot Product
Motivation: Work = Force* Distance
Box
FD
Fx
Fy
Box
DF
Fx
Fy
To find the work done in moving the box, we want the part of F in the direction of the distance
)cos(FFx
)cos( FDW
One interpretation of the dot product
)cos( DFDF
Where is the angle between F and D
A more useful definition
332211 babababa
)cos( 2222
bababa
You can show these two definitions are equal by considering the following triangle and applying the law of cosines! See page 808 for details
y
x
z
ba-b
aThink, what is |a|2?
Example
a=<2,-1,0>, b=<1,-8,-3>
Find a.b and the angle between a and b
The Dot Product
If a = <a1,a2,a3> and b=<b1,b2,b3> then
The dot product of a and b is a NUMB3R given by
332211 bababa ba
)cos(baba
The Dot Product
baba
)cos( 0)2/cos(
0)cos(2
0
a and b are orthogonal if and only if the dot product of a and b is 0
Other Remarks:
a
b
0)cos(2
Properties of the dot productSuppose a, b, and c are vectors and c is a scalar
1. a.a=|a|2
2. a.b=b.a3. a.(b+c) = (a.b)+(a.c)4. (ca).b=c(a.b)=a.(cb)5. 0.a=0
Yet another use of the dot product: Projections
a.b=|a| |b| cos( )
Think of our work example: this is ‘how much’ of b is in the direction of a
b
a|b| cos( )
We call this quantity the scalar projection of b on a
)cos(bababcompa
Think of it this way: The scalar projection is the length of the shadow of b cast upon a by a light directly above a
Q. How do we get the vector in the direction of a with length compab?
A.We need to multiply the unit vector in the direction of a by compab.
aaba
aa
ababproja
2
We call this the vector projection of b onto a
Examples/Practice!
Key Points
• Vector algebra: addition, subtraction, scalar multiplication
• Geometric interpretation• Unit vectors• The dot product and the angle between
vectors• Projections (algebraic and geometric)