Determine the equation of the line that is perpendicular to the lines r(t) equalsleft angle6t,1 plus3 t,4tright angle and R(s)equalsleft angle 2 pluss,negative 8 plus 3s,negative 12 plus 4 sright angle and passes through the point of intersection of the lines r and R, where tau equals 0 corresponds to the point of intersection.

Take the derivatives of each to get the tangent vectors:

Take the cross product of the tangent vectors to get a vector that is normal to both lines:

The two given lines intersect when :

that is, at the point (6, 4, 4).

The line perpendicular to both of the given lines through the origin is obtained by scaling the normal vector found earlier by ; translate this line by adding the vector to get the line we want,

The answer is True they are called groups and families