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The Cartesian equation of a line passing through a point (x1, y1, z1) and having directional ratios proportional to a,b,c is given by,

The required line passes through the point (2,1,3), now we need to find the direction ratios of the line which are a,b,c . this equation of the required line is,

We are given with the Cartesian equation of the two lines which are perpendicular to the given equation, as the lines are perpendicular with the required line so the dot product will result in zero.

The first line is , the dot product equation is,

a×1 + b×2 + c×3 = 0

a+2b+3c = 0 ………(i).

The second line is , the dot product equation is,

a×(–3) + b×2 + c×5 = 0

–3a+2b+5c = 0 ……….(ii).

Solving equations (i) and (ii), we get, by cross multiplication method,

= λ

= λ

= λ

a = 2λ, b = –7λ, c = 4λ

using a,b,c in the required equation we get,

This is the required equation.

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