L ( ) C c δ A a ( ) Vector - - Applications of Vector Algebra If the given lines intersect, then there must be a common point. Therefore, for some s t , ∈ , we have ( , , ) , , ) s s s Equating the coordinates of , x y and z we get , s s −= = and s = − . Solving the first two of the above three equations, we get s = and t = . These values of s and t satisfy the third equation.
So, the lines are intersecting. Now, using the value of s in ( ) or the value of t in ( ), the point of intersection ( , , ) of these two straight lines is obtained. If we take = + , then is a vector perpendicular to both the given straight lines. Therefore, the required straight line passing through ( , , ) and perpendicular to both the given straight lines is the same as the straight line passing through ( , , ) and parallel to .
Thus, the equation of the required straight line is ( ( ), m i m ∈ . Example . Determine whether the pair of straight lines ( ) ( ) ( ) ) s i are parallel. Find the shortest distance between them.
Comparing the given two equations with sb and sd we have , , , k b k c k d = + Clearly, b is not a scalar multiple of d . So, the two vectors are not parallel and hence the two lines are not parallel. The shortest distance between the two straight lines is given by δ = | ( ) ( ) | Now, b So, ( ) ( ( ) ( Therefore, the distance between the two given straight lines is zero.Thus, the given lines intersect each other. Vector - - Example .
Find the shortest distance between the two