Final answer:
The task is to find the parametric representation of the line of intersection for two planes and then minimize the distance from this line to a point. We begin by solving the system of equations to find the direction vector of the line, then express the line using parametric equations and finally set up an optimization problem.
Step-by-step explanation:
The problem requires us to find a parametric representation of the line of intersection of two planes: (x+y+z=1) and (x-2y-z=1) that passes through the point ((1,0,0)).To begin with, we can solve this system of equations to find a point and the direction vector for the line of intersection. Upon solving we find that the direction vector can be given by the cross product of the normals of the two planes. The normals are (1,1,1) for the first plane and (1,-2,-1) for the second plane. The cross product of these two vectors yields the direction vector for the line.
We can then write down the parametric equations for the line by using the point given (1,0,0) and the direction vector we obtained. Now we can express x, y, and z in terms of a single parameter, let's call it t. The resulting parametric equations can be used to determine any point on the line. To minimize the distance from the line to the origin, we can set up an optimization problem involving the distance formula derived from the pythagorean theorem.