Math

# Find the shortest distance (in units) between the lines 2y= -2x- 4 and -3y=3x-6

Step 1. Expressing the first straight line in terms of direction cosines

The given straight line is

The equations in terms of direction cosines be

Step 2. Expressing the second straight line in the symmetrical form

The given straight line is

Let the direction ratios of the above straight line be .

Since the straight line is perpendicular to the normals of both the planes, we have

By cross-multiplication, we get

Hence the direction ratios of the straight line are .

Let us take . Then,

Solving, we get .

Therefore the point on the straight line is .

Hence the equations of the straight line are

Here the direction cosines of the above line are

.

So the line can be rewritten as,

Step 3. Applying distance formula to find the shortest distance between the two given lines

We have found two straight lines

Applying the shortest distance formula between two straight lines, we get the required distance as,

units

units

units [ Expanding along the first row ]

units

units

units

units

units

The shortest distance is  units.

Formula to find the shortest distance between to skew lines:

Let two skew lines be

where  and  are direction cosines.

The shortest distance formula is

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