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Transcript
Multivariable Calculus
Summary 1-Vectors
DEFINITION: An n-tuple or an n-dimensional vector, is a symbol of the form
x1 , x2 , x3 , . . ., x
.
n
DEFINITION: R is the set of all n-tuples of real numbers.
2
Example: R is the set of all ordered pairs of real numbers. It is
represented by the Cartesian plane.
3
R is the set of all ordered triples of real numbers. It is
called the 3-dimensional vector space.
Physical concept of a vector: a vector is a directed arrow that begins at some initial point
called the tail and ends at some terminal point called the head.
2
Vectors in standard position or position vectors in R . An ordered pair

numbers represents a vector v
a, b
of real
from the origin (0, 0) to the point (a, b). a and b are called the
components of the vector.
3
Position Vectors in R . An ordered triple
a, b, c
of real numbers represents a vector from
the origin (0, 0, 0) to the point (a, b, c).
n
n
Vector in R . An n-tuple of real numbers represents a vector in R .
Definition: The norm of a vector

| v |=
x12  x 22  x32  .... x 2n
2
Note: In R
If
If

v = x 1, x 2 , x 3 , . . ., xn  is defined by
and R3 , the norm of a vector is given by the length of the vector.


v = a, b then its length of norm is given by | v |= a 2  b 2


v = a, b, c then its length of norm is given by | v |= a 2  b 2  c 2
Definition: A unit vector is a vector of length 1 or norm 1. To find the unit vector in the
1

v , multiply the vector by the scalar 
v
 1 

vector, then u =  v is a unit vector in the direction of v
v
direction of a give vector
-1-
. Therefore, if

v
is any
Operations of vector in
Addition:
a1 , b1 , c1  a2 , b2 , c2  a1  a2 , b1  b2 , c1  c2
Addition of vectors follows the parallelogram law of addition
a1 , b1 , c1  a2 , b2 , c2  a1  a2 , b1  b2 , c1  c2

Scalar Multiplication: if k is a scalar and v  a, b, c is a vector, then the scalar


multiplication k v is given by the vector k v  ka, kb, kc


Note: k v is a vector parallel to v in the same direction if k>0, in the opposite direction if k<0.


The length of k v is equal to the length of v multiply by |k|
  

Note: v1  v2  v1  (1v2 )




Note: Vectors u and v are parallel if an only if u  kv where k is a scalar.
2
Unit vectors in R : i  1, 0 and j  0, 1
Subtraction:
Unit vectors in
R 3 : i  1, 0, 0 and j  0, 1, 0 and k  0, 0, 1
2
Direction of a vector in R : The direction of a vector

v  a, b
can be indicated by the
b
, and  is the angle in standard position.
a

Theorem: If  (same as  ) and  are the angles that the vector v  a, b

x-axis and y-axis respectively, then the unit vector in the direction of v

is given by the vector u  cos  , cos   i cos   j cos 
angle

where
tan  
makes with

R 3 : The direction of a vector v  a, b, c can be indicated by the

 ,  ,  that the vector v  a, b makes with x-axis, y-axis and z-axis
Direction of a vector in
angles
respectively.

v  a, b, c makes with x-axis, y-axis

and z-axis respectively, then the unit vector in the direction of v

is given by the vector u  cos  , cos  , cos   i cos   j cos   k cos 
Theorem: If
, , 
are the angles that the vector
2
Ways of expressing a vector in R :

v  a, b

2. Algebraic form: v  ai  bj
 
3. Polar form: v   v ,  
 
 
4. Trigonometric form: v  v i cos   j sin   or v  v i cos   j cos  
1. Vector form:
-2-
Ways of expressing a vector in
R3 :

v  a, b, c

2. Algebraic form: v  ai  bj  ck
 
3. Trigonometric form: v  v i cos   j cos   k cos  
Vector from points P1 to P2
1. Vector form:

The vector from
P1  ( x1 , y1 ) to P2  ( x2 , y2 ) is given by x2  x1 , y2  y1

The vector from
P1  ( x1 , y1 , z1 ) to P2  ( x2 , y2 , z2 ) is given by x2  x1 , y2  y1 , z2  z1
Distance formulas:

The distance between two points in

In
R 3 , distance
is given by
D
x  x    y
x  x    y  y   z  z 
R2
is given by
D
2
2
2
2
1
1
2
2
2
1
2
2
1
Midpoint formulas:

The midpoint formula in

The midpoint formula in
 x1  x2 y1  y2 
,

2
2


 x  x y  y2 z1  z2 
,
R 3 is given by M=  1 2 , 1

2
2
2 

R2
is given by M= 
Circles and spheres:
2
 In R the equation of a circle with center (h, k) and radius r is given by
x  h
2
 ( y  k )2  r 2
R 3 the equation of a sphere with center (h, k, L) and radius r is given by
x  h2  ( y  k )2  ( z  L)2  r 2
2
2
2
3
Note: In R , x  h  ( y  k )  r represents a right circular cylinder

In
Dot product, scalar product or inner product:
Definition:
a1 , b1  a2 , b2  a1a2  b1b2
a1 , b1 , c1  a2 , b2 , c2  a1a2  b1b2  c1c2
Note: the dot product of two vectors is not a vector, it is a scalar.
Theorem: If and

u
and

v are vectors in
then


  
u  v  u v cos , where is the angle betweenu and v
-3-
 y1 
2
Theorem: Two vectors are perpendicular if and only if their dot product is 0.
n
Note: the above definitions can be generalize to R
Properties of the dot product. If u, v and w are vectors and k is a scalar then
1. u  u  u , that is, u  u  u
2. u  u  0
3. u  v  0 if and only if   90 , that is, u and v are perpendicular (orthogonal)
vectors
4. u  v  0 if and only if   90, that is  is acute
5. u  v  0 if and only if 90    180, that is  is obtuse
6. u  v  v  u , dot product is commutative
7. u  v  w  u  v  u  w , dot product is distributive over addition
8. k u  v   ku   v  u  kv , associativity of scalar and dot product
2
Projection: The orthogonal projection of a vector A on a vector B, is given by the
component of A parallel to B.
B
A
C
D



Let u  AB, let v  AD, then the projection of u on v is given by

Pr ojv u  Comv uAC
Theorem: If u and v are vectors, then the projection of u on v is given by the
vector
u v
u v
2 v. The length of the projection is given by Pr ojv u 
v
v
The vector w= u - Pr ojv u is called the component of u orthogonal to v.
Pr ojv u 
-4-