HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES

SHORTEST DISTANCE BETWEEN TWO LINES

1st of all we shall find out shortest distance between two Parallel lines.

Problem 1

Consider two parallel lines whose equations in vector form are given by
HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES

 Now comparing these equations with standard form , and write

 , and  vectors ,we get
HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES

Now applying this formula to find the shortest distance between two lines .
HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES
As it is clear from formula , we have to find cross product of  and  and then magnitude of vector 

HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES

Now find the magnitude of  × vector
HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES
                                 =√(81)+(196)+(16)
                                 =√293
Magnitude of 
HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES
                    = √49
           = 7
Putting all these values in  the formula of Shortest Distance between two lines .
HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES
Now Find distance between two skew lines i.e. Lines which are not Parallel lines.

Problem 2

Consider two parallel lines whose equations in vector form are given by
 Now comparing these equations with standard form , and write

 , and  vectors ,we get
 Now applying this formula to find the shortest distance between two lines 

 Now find cross product and then  magnitude of these two vectors


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