Minimum Distance of a Point from a Plane

To find the minimum distance of a point (1, 1, 1) from the plane x + y + z = 1, we can use the formula for the distance between a point and a plane. However, in this case, we need to measure the distance perpendicular to the line x - x1/1 = y - y1/2 = z - z1/3. Let's break down the solution into the following steps:

Step 1: Find the equation of the given plane
The equation of the plane is x + y + z = 1. This equation can also be written in the form Ax + By + Cz + D = 0, where A = 1, B = 1, C = 1, and D = -1.

Step 2: Find the direction ratios of the given line
The direction ratios of the line x - x1/1 = y - y1/2 = z - z1/3 can be obtained by comparing the coefficients of x, y, and z. In this case, the direction ratios are 1, 2, and 3, respectively.

Step 3: Find the direction ratios of the normal to the plane
Since the distance needs to be measured perpendicular to the line, we need to find the direction ratios of the normal to the plane. The normal to the plane is given by the coefficients of x, y, and z in the equation of the plane. In this case, the direction ratios of the normal are 1, 1, and 1.

Step 4: Find the dot product of the direction ratios
To measure the distance perpendicular to the line, we need to find the dot product of the direction ratios of the normal to the plane and the line. The dot product can be calculated using the formula a1b1 + a2b2 + a3b3, where (a1, a2, a3) are the direction ratios of the normal and (b1, b2, b3) are the direction ratios of the line. In this case, the dot product is (1)(1) + (1)(2) + (1)(3) = 1 + 2 + 3 = 6.

Step 5: Calculate the minimum distance
The minimum distance between the point (1, 1, 1) and the plane can be calculated using the formula: distance = |Ax1 + By1 + Cz1 + D| / √(A^2 + B^2 + C^2). In this case, the coordinates of the point are (x1, y1, z1) = (1, 1, 1) and the coefficients of the plane are A = 1, B = 1, C = 1, and D = -1. Plugging in these values, the minimum distance is |(1)(1) + (1)(1) + (1)(1) - 1| / √(1^2 + 1^2 + 1^2) = 0 / √3 = 0.

Conclusion:
The minimum distance of the point (1, 1,