Question

to find perpendicular distance of the line x+a/=y+b/m=z+c/n

Answer 1

Let there be a point P(lh-a, mh-b, nh-c),

So, OP vector is now perpendicular to a vector along line, therefore, OP. (li+mj+nk)=0

solve this to get 'h' and so you will get point P.

Then apply distance formula to get the length OP

Answer 2

Let the given line be L. Point P(-a, -b, -c) lies on it. Direction ratios of L are: (l, m, n). Origin be O(0,0,0). We want perpendicular distance d from O onto L.

Direction ratios of OP are: (a, b, c).
d = OP Sinθ. θ is the angle between OP and L. It is obtained by vector cross product.

$\vec{L}= l \hat{i}+ m \hat{j} + n\hat{k} \\ d = OP * Sin \theta = \frac { | OP \times \vec{L} | }{ | \vec{L} | }$ $=( | \hatIi} (bn-cm) - \hat{j} (an - cl) + \hat{k} (am-bl) | )\\ / (\sqrt{l^2+m^2+ n^2)$ $d^2=\frac{ (bn-cm)^2+(an-cl)^2+(am-bl)^2} {l^2+m^2+n^2 }$

we could use the matrix method for obtaining the cross product of vectors.
| i j k |
| a b c |
| l m n |