Let I be the image of the point P(2, 0, 1) in the line x-1 2 = y+1 1 = z-2 3 . If O is the origin, then the square of the area of the parallelogram whose adjacent sides are represented by the vectors OP and OI is :

Step-by-step solution

Let the given line be x-1 2 = y+1 1 = z-2 3 = . A general point F on this line is F(2 + 1, - 1, 3 + 2) . Assuming F is the foot of the perpendicular from P(2, 0, 1) to the line, the direction ratios of PF are: 2 + 1 - 2, - 1 - 0, 3 + 2 - 1 = 2 - 1, - 1, 3 + 1 Since PF is perpendicular to the line, the dot product of their direction ratios is zero: 2(2 - 1) + 1( - 1) + 3(3 + 1) = 0 4 - 2 + - 1 + 9 + 3 = 0 14 = 0 = 0 Thus, the foot of the perpendicular is F(1, -1, 2) . Since F is the midpoint of P and its image I(x,