Three dimensional Geometry

i ,j and k 3 unit vectors in direction of x axis y axis and z axis

Direction cosine give the angle with which align makes an angle with the axis formula for the direction cosine is

l = a cos ¢

m = b cos ¢ 

n = c cos ¢ 

Where a b and c are the direction ratios

 

Direction cosines and direction ratios of a vector

Consider the position vector of a point P(x, y, z) The angles α, β, γ made by the vector  with the positive directions of x, y and z-axes respectively,are called its direction angles. The cosine values of these angles, i.e., cos α, cos β and cos γ are called direction cosines of the vector , and usually denoted by l, m and n, respectively.

i.e. l = cosα, m = cosβ and n = cosγ. The direction of a line cannot be fixed in space by knowing anyone or any two angles.

one may note that the triangle OAP is right angled, and in it, we have l = cosα=x/r, Similarly, from the right angled triangles OBP and OCP, we may write  m = cosβ=y/r and

  • n = cosγ=z/r
  • x=lr,y=mr,z=nr.

Thus, the coordinates of the point P may also be expressed as (lr, mr,nr). The numbers lr, mr and nr, proportional to the direction cosines are called as direction ratios of vector , and denoted as a, b and c, respectively. Where r  denotes the magnitude of the vector and it is given by,

Example : Find the direction ratios and direction cosines of a line joining the points (3, -4, 6) and (5, 2, 5).

Solution:

Given points are A(3, -4, 6) and B(5, 2, 5)

The direction ratios of the line joining AB is

a = x– x1 = 5 – 3 = 2

b = y– y1 = 2 + 4 = 6

c = z– z1 = 5 – 6 = -1

AB =

So direction cosines of the line = 2/√41, 6/√41, -1/√41.

Example : Find the direction cosines of the line joining the points (2,1,2) and (4,2,0).

Solution:

Let the points are A(2,1,2) and B(4,2,0).

x2-x1 = 4-2 = 2

y2-y1 = 2-1 = 1

z2-z1 = 0-2 = -2

AB = √(22+12+(-2)2)= 3

Hence the direction cosines are ⅔, ⅓, -⅔.

Example : Find the direction cosines of the line joining the points (2,3,-1) and (3,-2,1).

Solution:

Let the points are A(2,3,-1) and B(3,-2,1).

x2-x1 = 3-2 = 1

y2-y1 = -2-3 = -5

z2-z1 = 1- (-1) = 2

AB = √(12+(-5)2+22)= √30

Hence the direction cosines are 1/√30, -5/√30, 2/√30.