Find the shortest distance between the lines given by and
Given two lines,
Taking equation (i),
We know, the vector equation of a line passing through a point and parallel to a vector is where
= Position vector of the point through which line passes
= Normal to the line
Comparing this with equation (iii), we get
Now take equation (ii)
Similarly from (iv)
So, the shortest distance between two lines can be represented as:
solve
Taking 1st row and 1st column, we multiply the 1st element of the row (a??) with the difference of the product of the opposite elements , excluding the 1st row and the 1st column;
Here
Now, we take the 2nd column and 1st row, and multiply the 2nd element of the row (a??) with the difference of the product of opposite elements (a?? x a?? - a?? x a??)
Here
Finally, taking the 1st row and 3rd column , we multiply the 3rd element of the row (a??) with the difference of the product of opposite elements (a?? x a?? - a?? x a??), excluding the 1st row and 3rd column.
Here
Further simplifying it.
And,
Substituting the values from (v), (vi) and (vii) in d, we get
Thus, the shortest distance between the lines is 14 units.