Find the shortest distance from point P 2 4 6 to a point on

Find the shortest distance from point P = (-2, 4, 6) to a point on line given by l : (x, y, z) = (4t, 2t, 4t). The distance is

Solution

PQ×u D = -------- u where P is the point P(-2,4,6) and Q is any point on the line, say when t=0 x=4t, y=2t, z=4t we have the point Q(0,0,0), and u = < 4, 2, 4>, the direction vector for the line. We calculate the vector PQ = < -2-0, 4-(0), 6-(0) > = < -2, 4, 6 > We find the cross product: PQ×u = < -2, 4, 6 >×<4 ,2 , 4> PQ×u = 28i + 32j - 20k We find the norms: PQ×u = sqrt [28²+32²+(-20)²]= = sqrt(2208) u = sqrt 4²+(2)²+4² = sqrt(36) PQ×u sqrt[2208] D = -------- = ------ =7.8 u sqrt36