Math 224
–
Calculus III
1
12.2 Vectors
Recommended Homework: # 135(odds)
Often quantities in science have both a
magnitude
and a
direction
associated with them:
Example:
Velocity
Example:
Force
In order to handle this situation we describe these quantities using
vectors
:
Picture & notation
:
Notes
:
To indicate a vector we write:
v
or
v
We can also indicate a vector by its start and end points:
v
AB
The magnitude is represented by the length of the vector
The magnitude (length) of the vector
v
AB
is denoted
v
or
AB
or
AB
Two vectors are equivalent (or equal) if they have the same direction and the same magnitude.
They DO NOT need to have the same position:
v
AB
CD
w
The zero vector
0
is the vector with no length and no direction (or any direction!)
A
B
v
C
D
w
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Math 224
–
Calculus III
2
Vector Addition
Consider the two vectors
u
and
v
:
There are two ways (graphically) we can understand
u+ v
:
Triangle Law
Parallelogram Law
Note that vector addition is commutative:
u+v = v
u
Scalar Multiplication
Multiplication of a vector
by a constant (“scalar”)
c
The length of the vector
c
v
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 Fall '09
 Harris
 Calculus, Linear Algebra, Vectors, #, Standard basis, 3j, AB CD

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