## 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

**|**

**×**

**|**

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**

The magnitude of this vector

= √(169+64+4)=√(237)

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

= -48 -10

= -58

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

SD = 58

**/(**√237) Units## 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

## 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.

After solving this determinant and simplification we get ,

The magnitude of this vector is

= √2069

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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