Elementary Vector Analysis
In order to measure many physical quantities, such as force or
velocity, we need to determine both a magnitude and a direction. Such
quantities are conveniently represented as vectors.
The direction of a vector v in 3space is specified by its
components in the x, y, and z directions, respectively:
(x,y,z) or xi + yj + zk,
i = (1,0,0)
j = (0,1,0)
k = (0,0,1)

where i, j, and k are the coordinate vectors
along the x, y, and zaxes.
The magnitude of a vector v = (x,y,z), also called its
length or norm, is given by
 
v 
 =  √ 
x^{2}+y^{2}+z^{2}



Notes
 Vectors can be defined in any number of dimensions, though we
focus here only on 3space.
 When drawing a vector in 3space, where you position the vector
is unimportant; the vector's essential properties are just its
magnitude and its direction. Two vectors are equal if and only
if corresponding components are equal.
 A vector of norm 1 is called a unit vector. The
coordinate vectors are examples of unit vectors.
 The zero vector, 0 = (0,0,0), is the only vector with
magnitude 0.
Basic Operations on Vectors
To add or subtract vectors
u = (u_{1},u_{2},u_{3}) and
v = (v_{1},v_{2},v_{3}),
add or subtract the corresponding coordinates:
u + v = (u_{1}+v_{1},u_{2}+v_{2},u_{3}+v_{3})
u  v = (u_{1}v_{1},u_{2}v_{2},u_{3}v_{3})
To mulitply vector u by a scalar k, multiply each coordinate
of u by k:
k u = (ku_{1},ku_{2},ku_{3})
Example
The vector v = (2,1,2) = 2i + j  2k
has magnitude
 
v 
 = 
 ___________ √2^{2} +1^{2} (2)^{2}

= 3. 

Thus, the vector ^{1}/_{3}v =
(^{2}/_{3},^{1}/_{3},^{2}/_{3}) is a unit vector in the same direction as v.
In general, for v ≠
0, we can scale (or normalize)
v to the unit vector v/ v
pointing in the same direction as v.
Le componenti del versore (vettore unitario) u ottenuto normalizzando v = (v_{1},v_{2},v_{3}) vengono dette anche coseni direttori
(direction cosines) di v in quanto sono dati dal coseno degli angoli che v forma con i tre assi coordinati:
u_{i} = cos(∠ v x_{i}) (avendo indicato gli assi x, y e z con x_{1},
x_{2} e x_{3}).
La figura a lato illustra il caso di u_{3}. Con un'analoga costruzione si possono illustrare gli altri due casi.
Si^{ }noti che posso scrivere sia cos(∠ v x_{i}) che cos(∠ x_{i} v) in quanto cos(α) = cos(–α).
 
Una^{ }somma del tipo
a_{1}v_{1}+a_{2}v_{2}+…+a_{n}v_{n}
(con a_{i} numeri reali e v_{i} vettori) viene detta combinazione lineare
dei vettori v_{1}, v_{2}, …, v_{n}.
Ogni vettore può essere espresso (in modo unico) come combinazione lineare di i, j e k.

Dot Product (prodotto scalare)
Let u = (u_{1},u_{2},u_{3})
and v = (v_{1},v_{2},v_{3}).
The dot product u · v
(also called the scalar procuct or Euclidean inner
product) of u and v is defined in two distinct
(though equivalent) ways:



u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3} 







where 0 ≤ θ ≤ π is the angle between 
u 
and 
v 


 
 
Why are the two definitions equivalent?
Properties of the Dot Product
 u · v = v · u
 u · (v + w) = (u · v) + (u · w)
 u · u =  u ^{2}
See if you can verify each of these!
Example
If u = (1,2,2) and v = (4,0,2), then
Using the second definition of the dot product with  u  = 3 and  v  = 2√5,
so cosθ = 0, yielding θ = π/2.
Though we might not have guessed it, u and v are
perpendicular to each other!
In general,
Two nonzero vectors u and v are
perpendicular (or orthogonal) if and only if
u ·v = 0.
Proof
Projection of a Vector
It is often useful to resolve a vector v into the sum of
vector components parallel and perpendicular to a vector u.
Consider first the parallel component, which is called the
projection of v onto u. This projection should be in
the direction of u and should have magnitude vcosθ, where 0 ≤ θ ≤ π is the angle between
u and v. Let's normalize u to
u/ u  and then scale this by
the magnitude  v cosθ:
Projection of v onto u



= 
v
u
cosθ
u^{2}

u 


The perpendicular vector component of v is then just the
difference between v and the projection of v onto
u.
In summary,
projection of v onto u: 

vector component of v
perpendicular to u:



Cross Product (prodotto vettoriale)
Let u = (u_{1},u_{2},u_{3}) and v = (v_{1},v_{2},v_{3}). The cross product u × v yields a vector perpendicular to both u and v
with direction determined by the righthand rule. Specifically,
u × v =
(u_{2}v_{3}u_{3}v_{2})i 
(u_{1}v_{3}u_{3}v_{1})j +
(u_{1}v_{2}u_{2}v_{1})k
or: u × v = 
     

i   j   k 
      
u_{x}  u_{y}  u_{z} 
v_{x}  v_{y}  v_{z} 
It can also be shown that
 u × v
 =
 u
 
v
 sinθ
for u ≠ 0,
for v ≠ 0
where 0 ≤
θ
≤
π
is the angle between
u and v.
Proof
Thus, the magnitude  u ×v  gives
the area of the parallelogram formed by u and v.
As implied by the geometric interpretation,
Non zero vectors u and v are
parallel if and only if u × v = 0.
Proof
Properties of the Cross Product
 u × v =  ( v × u)
 u × ( v + w ) = (u × v ) + ( u × w )
 u × u = 0
Again, see if you can verify each of these.
Connections between the dot product and cross product (Lagrange's Identity)
Key Concepts [index]
Let u = (u_{1},u_{2},u_{3}) and v = (v_{1},v_{2},v_{3}).
 Basic Operations, Norm of a vector



(u_{1}+v_{1},u_{2}+v_{2},u_{3}+v_{3}) 



(u_{1}v_{1},u_{2}v_{2},u_{3}v_{3}) 







 

 Dot Product



u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3} 






where 0 ≤ θ
≤
π
is the angle between u and v 

 

For u ≠
0, v ≠
0, u · v = 0 if and only if u is
orthogonal to v.
 Projection of a Vector
projection of v onto u: 

vector component of v
perpendicular to u:


 Cross Product
u × v =
(u_{2}v_{3}u_{3}v_{2})i 
(u_{1}v_{3}u_{3}v_{1})j +
(u_{1}v_{2}u_{2}v_{1})k
 u × v
 =
 u
  v
 sinθ
for u ≠ 0,
for v ≠ 0
where 0 ≤
θ
≤
π is the angle between
u and v.
For u ≠
0, v ≠
0, u × v = 0 if and only if u is
parallel to v.
