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Home » GATE Study Material » Engineering sciences » Elementary Vector Analysis

Elementary Vector Analysis

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Elementary Vector Analysis

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 3-space 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 z-axes.

The magnitude of a vector v = (x,y,z), also called its length or norm, is given by

|| v || =
x2+y2+z2
 
 

 

Notes

 

  • Vectors can be defined in any number of dimensions, though we focus here only on 3-space.
  • drawing a vector in 3-space, 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 = (u1,u2,u3) and v = (v1,v2,v3),

add or subtract the corresponding coordinates:

 

u + v = (u1+v1,u2+v2,u3+v3)

u - v = (u1-v1,u2-v2,u3-v3)


To mulitply vector u by a scalar k, multiply each coordinate of u by k:

 

k u = (ku1,ku2,ku3)

 

Example

The vector v = (2,1,-2) = 2i + j - 2k has magnitude

|| v || =     ___________
22 +12 -(-2)2
 
= 3.

Thus, the vector 1/3v = (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.

 

Dot Product

Let u = (u1,u2,u3) and v = (v1,v2,v3). The dot product uv (also called the scalar procuct or Euclidean inner product) of u and v is defined in two distinct (though equivalent) ways:

 

uv= u1v1+u2v2+u3v3 uv is a number �

� ||
u || || v || cosq if u 0, v 0 0 if u = 0 or v = 0 where 0 q p is the angle between u and v

 

Properties of the Dot Product

 

  • uv = vu

     

  • u � (v + w) = (uv) + (uw)

     

  • uu = || u ||2

     

See if you can verify each of these!

 

Example


If u = (1,-2,2) and v = (-4,0,2), then

uv
=
(1)(-4)+(-2)(0)+(2)(2)
 
=
-4+0+4
 
=
0
 

Using the second definition of the dot product with || u || = 3 and || v || = 25,

  uv = 0 = 65cosq
so cosq = 0, yielding q = p/2.

Though we might not have guessed it, u and v are perpendicular to each other!

In general,

 

Two non-zero vectors u and v are perpendicular (or orthogonal) if and only if uv = 0.

 

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 || v||cosq, where 0 q p is the angle between u and v. Let's normalize u to u/|| u || and then scale this by the magnitude || v ||cosq:

 

Projection of v onto u
= (|| v || cosq) u
||u||
 
= ||v|| ||u|| cosq
||u||2
u
= vu
||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:
= vu
||u||2
u
vector component of v
perpendicular to u:
= v - vu
||u||2
u
 

 

Cross Product

Let u = (u1,u2,u3) and v = (v1,v2,v3). The cross product uv yields a vector perpendicular to both u and v with direction determined by the right-hand rule. Specifically,

 

uv is a vector
uv = (u2v3-u3v2)i - (u1v3-u3v1)j + (u1v2-u2v1)k

It can also be shown that

 

|| uv || = || u || || v || sinq for u 0, for v 0

where 0 q p is the angle between u and v.



Thus, the magnitude || uv || 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 uv = 0.

 

Properties of the Cross Product

 

  • uv = - ( vu)

     

  • u � ( v + w ) = (uv ) + ( uw )

     

  • uu = 0

     

Again, see if you can verify each of these.

In the following Exploration, select values for the components of u and v. You will see uv and uv computed and u, v, and uv displayed on a coordinate system.

 

Exploration

 

Key Concepts

Let u = (u1,u2,u3) and v = (v1,v2,v3).

 

  • Basic Operations, Norm of a vector

     

     
      u + v  
    =
    (u1+v1,u2+v2,u3+v3)
      u - v  
    =
    (u1-v1,u2-v2,u3-v3)
    k u  
    =
    (ku1,ku2,ku3)
    || v ||
    =
     
    x2+y2+z2
     
     
     

     

  • Dot Product

    uv

    nowrap="nowrap">u1v1+u2v2+u3v3 =

    || u|| ||vvcosqif u 0, v 0 0u = 0 or v = 0

    where 0 q p is the angle between u and v

    For u 0, v 0, uv = 0 if and only if u is orthogonal to v.

     

  • Projection of a Vector

     

    projection of v onto u:
    = vu
    ||u||2
    u
    vector component of v
    perpendicular to u:
    = v - vu
    ||u||2
    u
     

  • Cross Product

     

    uv = (u2v3-u3v2)i - (u1v3-u3v1)j + (u1v2-u2v1)k

    || uv || = || u || || v || sinq for u 0, for v 0

    where 0 q p is the angle between u and v.

    For u 0, v 0, uv = 0 if and only if u is parallel to v.



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