Show that the lines `(x-4)/(1)=(y+3)/(-4)=(z+1)/7 and (x-1)/(2) =(y+1)/(-3)=(z+10)/(8)` intersect. Find the coordinates of their point of intersection.

Explanation

Let `(x-4)/(1) = (y+3)/(-4) = (z+1)/(7) = lambda`
Co-ordinates of any point on A this line.
`A(lambda |4, 4 lambda, 4lambda" "3, 7lambda-1)`
Again Let, `(x-1)/(2) = (y+1)/(-3) = (z+10)/(8) = mu`
Co-ordinates of any point B on this line.
`B(2mu+1,-3mu-1,8mu-10)`
If these lines intersect, one point will be common.
If A and B coincide then
`lambda + 4 = 2 mu + 1 rArr lambda = 2mu-3"......"(1)`
`-4lambda-3=-3mu-1 rArr 4 lambda=3mu-2"........."(2)`
`7lambda-1=8mu-10-10 rArr 7lambda=8mu-9"........."(3)`
From eq. (1) and (2)
`lambda = 1 " " mu = 2`
Eq. (3) satisfies with these values.
Therefore given lines intersect.
Point of intersection `= A(lambda+4,-4lambda-3,7lambda-1)`
`= A (5,-7,6)`