3D Geometry - 2

Illustration: Find angle b/w lines whose direction cosines are   &   .

Ans:   Let be angle
         cos = l₁l₂ + m₁m₂ + n₁n₂
                
         cos = -   = 120⁰


Straight line:

Vectorial eq. of line:

Let a line passing through a point of P.V. & || to given line EF().
Then, eq. of line  

Proof: Let P be any point on line AP & its P.V. is
Then, = =   (By law)
        

Dumb Question: Why = ?

Ans: Since is || to EF of P.V.
So, =

Note: Eq. of line through origin & || to is =


Cartesion eq. of straight line:

Passing through point & parallel to given vector .
Let line is passing through A(x₁, y₁, z₁) & || to EF whose d.r.'s are a, b, c.


d.r.'s of AP = (x - x₁, y - y₁, z - z₁)
d.r.'s of EF = (a, b, c)
Since EF || AP, then,
   b Eq. of line in parametric form.


Illustration: Find eq. of line || to & passing through pt.(1, 2, 3) ?

Ans: A(1, 2, 3)   =
eq. of line passing through & || to
    = () + ()


Vector eq. of line passing through two given points:

Vector eq. of line passing through two points whose P.V. S   & is:
     = + ( - )

Proof: is collinear with


  =



Cartesian form:

Eq. of line passing through (x₁, y₁, z₁) & (x₂, y², z₂)


D.R.'s of AB = (x₂ - x₁, y₂ - y₁, z₂ - z₁)
D.R.'s of AP = (x - x₁, y - y₁, z - z₁)
Since AB || AP
    


Illustration: Find cartesian eq. of line are  6x + 2 = 3y - 1 = 2z + 2. Find its direction ratios.

Ans:     is cartesion eq. of line
         6x + 2 = 3y - 1 = 2z + 2
         6(x + ) = 3(y - ) = 2(z + 1)
        
         on comparing   a = , b = , c =


Angle b/w Two lines:

    
     =
Angle b/w two lines   cos =
If  
       =
    


Condition for perpendicularity:

     . = 0   a₁a₂ + b₁b₂ + c₁c₂


Condition for parallelism:


Dumb Question: Why angle between pair of lines
      

Ans:     &  
        


Perpendicular distance of point froma line:

(a) Castesian form:


    
Suppose 'L' is foot of line of .
Since L lies on line AB so,
coordinate of L is
     =
i.e. L(x₁ + a, y₁ + b, z₁ + c)
direction ratios of PL are:
     (x₁ + a - , y₁ + b - , z₁ + c - )
also direction ratios of AB are (a, b, c)
Since PL AB
a(x₁ + a - ) + b(b - ) + c(z₁ + c - ) = 0