HOW TO FIND THE PERPENDICULAR DISTANCE BETWEEN TWO SKEW LINES AND PARALLEL LINES

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The Shortest Distance Between Skew Lines

The shortest distance between the lines is the distance which is perpendicular to both the lines given as compared to any other lines that joins these two skew lines.

Vector Form

We shall consider two skew lines  L1 and L2 and  we are to calculate the distance between them. The equations of the given lines are:

Here  vector  and vector  are the vectors through which line (1) and (2) passes and and   are the vectors which are parallel to lines    L1 and L2  respectively. 

Then perform the following steps

(1)                 Calculate  -  

(2 )                Calculate   ×

(3)                Calculate    |  ×  | 

(4 )                Calculate ( − ) . (   ×  )

and if the dot product of these two vectors come out to be negative then take its absolute value as distance can not be a negative quantity  .

(5 )                Put all these values in the formula given below and  the value so calculated is the shortest distance between two skew Lines.

 ⃗⃗

Problem : How to find  the shortest distance between two skew lines in vector form whose equations are given by

  Now write the values of    ,, and 

Now find out the difference of and 

After solving this determinant and little simplification we get ,

The magnitude of this vector 

                             = √(169+64+4)=√(237)

Now find Dot Product    ( -  )  and  (   ×  ) ,

                                                    = (0)(-13)+(-6)(8)+(5)(-2)

                                                    = -48 -10

                                                    = -58

Taking the absolute value of

|   (− )  .  (   ×  ) | = 58

Now putting all these values in SD Formula written above , we can have

SD = 58/(√237) Units

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The Shortest Distance Between Parallel Lines

Consider two parallel lines

Here  vector  and  vector  are the vectors through which line (1) and (2) passes and    is the vector which is parallel to both lines    L1 and L2  respectively. Now perform the following steps 

( 1 )                 Calculate  -  

( 2 )                Calculate    | | 

( 3 )                Calculate ( -  )     ×  ,

     

Put all these values in the formula given below and the value so calculated is the shortest distance between two Parallel Lines, and if it comes to be negative then take its absolute value as distance can not be negative

Problem 2 : How to find  the shortest distance between two Parallel lines in vector form whose equations are given by

As we know these are parallel line because both these equations are parallel to same vector  -2i +3J+5k.

Now write the values of    ,, and  as follows

Now find out the difference of and 

Now  Calculate ( -  )     ×  ,

     After solving this determinant and  simplification we get ,            

                  

 

The magnitude of this vector is   

= √2069

Similarly the magnitude of vector    is √38

  Put all these values in the formula given below and the value so calculated is the shortest distance between two Parallel Lines, and if it comes to be negative then take its absolute value as distance can not be negative

SD = √(2069 /38) Units

At Last

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