Find a vector equation for the line through the point P 1 1

Find a vector equation for the line through the point P = (-1, 1. -2) and parallel to the vector v = (2, -2, -4). Assume r(0) = - 1i + 1f - 2k and that v is the velocity vector of the line. r(t) =

Solution

Given vector v = ( 2, - 2, - 4) and r(0) = -i + j - 2k, then vector equation of line through P and parallel to v is given by

r(t) = r(0) + tv

   = (-i + j - 2 k ) + t ( 2, -2, -4)

   = (-i + j - 2 k ) + ( 2t, -2t, -4t)

     = (-i + j - 2 k ) + ( 2ti - 2tj -4tk )

r (t) = (2t - 1 ) i + ( 1 -2t ) j + ( -2 - 4t ) k