Q.Find the equation of the line that passes through the origin and (5, 2, 4).

A line through the origin and a given point has direction ratios equal to that point's coordinates.

A line passing through the origin $(0,0,0)$ and the point $(5,2,4)$ has direction ratios given directly by the coordinates of the second point (since the direction vector is the displacement from the origin):

$(5-0,\ 2-0,\ 4-0) = (5,2,4)$

Symmetric (cartesian) form of a line through $(x_0,y_0,z_0)$ with direction ratios $(a,b,c)$: …