Q.Find the equation of a straight line which passes through the point (1, 2, 3) and is perpendicular to both of the lines x/2 = y/1 = z/3 and (x-3)/-1 = (y-2)/3 = (z+5)/5.

A line perpendicular to two given lines must be along their cross product; find that cross product, then write the line through the given point with that direction.

The two given lines have direction ratios $\vec d_1 = (2,1,3)$ (from $x/2=y/1=z/3$) and $\vec d_2=(-1,3,5)$ (from $(x-3)/-1=(y-2)/3=(z+5)/5$).

A line perpendicular to both must be parallel to $\vec d_1\times\vec d_2$:

$\vec d_1\times\vec d_2 = \begin{vmatrix}\hat i&\hat j&\hat k\\2&1&3\\-1&3&5\end{vmatrix}$

$\hat i$: $(1)(5)-(3)(3) = 5-9=-4$

$\hat j$: $-[(2)(5)-(3)(-1)] = -(10+3)=-13$

…