Find the length and equations of the line of the shortest distance | Filo

Class 12

Math

3D Geometry

Three Dimensional Geometry

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150

Find the length and equations of the line of the shortest distance between the lines given by
and

Solution: Given lines are

The vector form of direction ratio's of line is

The vector form of direction ratio's of line is

As we know that

The line of shortest distance will be perpendicular to both lines so the direction ratio's of shortest distance line will be cross product of direction ratio's of both the lines

So direction of shortest distance is

Let

So the point of contact of first line and shortest distance line is

The point of contact of second line and shortest distance line is

As we know that

The direction ratio's of line joining and is

The direction ratio's of shortest distance line is

As we know that

If two lines having direction ratio's as and are perpendicular then

As line of shortest distance is perpendicular to both lines

Solving and gives

Substituting we get the common points as and

Distance between these two points will be the shortest distance

So shortest distance is

The equation of shortest distance line is
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