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Reducing Cartesian Form of a line to Vector Form and vice-versa

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Cartesian Form of a line passing through a given point

reducingcartesianform1

Equation of a line passing through a given point P(x1, y1, z1) and parallel to a given vector b having direction ratios <a, b, c> is

                reducingcartesianform2
If <l, m, n> are the direction cosines then its equation is given by
                reducingcartesianform3



Reduction of Cartesian form to the Vector form

If reducingcartesianform2  is the equation of line then equate this to a
    
constant, say λ to get the vector form.
So, reducingcartesianform2 =   λ
         
reducingcartesianform4
Hence the vector equation of line passing through a point with position vector a and parallel to b is r=a+λb

Reduction of Vector form to the Cartesian Form

reducingcartesianform5
              

                reducingcartesianform2 =    λ, which is the Cartesian form
                             

Cartesian equation of a line passing through two points

If P(x1, y1, z1) and Q(x2, y2, z2) are two given points then Cartesian equation of line is
reducingcartesianform6

Reduction of Cartesian form to Vector form

As in the previous case, equate the Cartesian form of the line to λ, so that it

  1. becomes   reducingcartesianform7

    Reduction of Vector form to Cartesian form

    reducingcartesianform8
    Hence from the above three equations, we can write

  2.       reducingcartesianform6  =     λ, which is the Cartesian form.
       
               

    Angle between two lines

    If reducingcartesianform9are the vector equations of two lines then angle between them is given by
                                   reducingcartesianform10
    Equations of two lines then angle between them is given by
                         reducingcartesianform12

    Condition for parallelism and perpendicularity

    •    If the lines are parallel then reducingcartesianform11
                                            
    •    If the lines are perpendicular then a1a2+b1b2+c1c2=0

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