Determine the parametric equations of the line through a point and orthogonal to 2 lines?

Let the direction ratios of the required line be (l, m, n)

As given, the direction ratios of Line 1 are (2, 3, -4) and that of
Line 2 are (3, 1, -5)

Since required line is orthogonal to lie 1 and 2, we have

2l + 3m - 4n = 0 and 3l + m - 5n = 0

Solving these two, l = 11n/7* and m = 2n/7
[*Initially on a typographic error, I mentioned this as 11m/7, which is now corrected as 11n/7 on review].

Hence, l:m:n = 11n/7 : 2n/7 : n = 11:2:7

So the dirction ratios of the line are (11, 2, 7)

It passes through (3, 1, 2)

Hence its equation in analytical form is:

[(x - 3)/11] = [(y - 1)/2] = [(z - 2)/7]

The same in parametric form is:
Any point on the line is: (3 + 11?, 1 + 2?,. 2 + 7?)

Additional Information:

Consider the two eqauitons, 2l + 3m - 4n = 0 and 3l + m - 5n = 0
as, 2l + 3m = 4n; 3l + m = 5n. Now solve these two and get the values of 'l' and 'm' in terms of 'n'.