We are given with the direction ratios and the vector equations of two lines, we have to find the angle between the two lines, and weather they are parallel or perpendicular to each other.

The direction ratios are a,b,c and b–c, c–a, a–b .

The vectors with these direction ratios are,

= a+b+c and = (b–c)+(c–a)+(a–b)

By using dot product equation to find the angle between them, we get,

cos θ =

cos θ =

cos θ =

cos θ =

cos θ = 0

θ =