Cartesian Form of a line passing through a given point

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

Reduction of Cartesian form to the Vector form

If   is the equation of line then equate this to a
    
constant, say λ to get the vector form.
So,  =   λ
         

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

                =    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

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  
   

   Reduction of Vector form to Cartesian form

   
   Hence from the above three equations, we can write
2. 
           =     λ, which is the Cartesian form.
      
              
   

   Angle between two lines

   If are the vector equations of two lines then angle between them is given by
                                 
   Equations of two lines then angle between them is given by
                       
   

   Condition for parallelism and perpendicularity

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

Now try it yourself!  Should you still need any help, click here to schedule live online session with e Tutor!

About eAge Tutoring:

eAgeTutor.com is the premium online tutoring provider.  Using materials developed by highly qualified educators and leading content developers, a team of top-notch software experts, and a group of passionate educators, eAgeTutor works to ensure the success and satisfaction of all of its students.  

Contact us today to learn more about our tutoring programs and discuss how we can help make the dreams of the student in your life come true!

Reference Links:

• http://en.wikipedia.org/wiki/Angle
• http://en.wikipedia.org/wiki/Direction_cosine
• http://en.wikipedia.org/wiki/Parallel_(geometry)
• http://en.wikipedia.org/wiki/Perpendicular