Find the distance from the point M (0; 1; 2) to the straight line (x-1) / 2 = y = (z-2) / 0

Find the distance from the point M (0; 1; 2) to the straight line (x-1) / 2 = y = (z-2) / 0 and the projection of the point M onto the straight line.

1. Distance from point M to any point N lying on a given straight line:

{y = (x – 1) / 2;
{z – 2 = 0;
{y = 1/2 * x – 1/2;
{z = 2.
The straight line lies in the plane z = 2.

M (0; 1; 2).

MN ^ 2 = (x – 0) ^ 2 + (y – 1) ^ 2 + (z – 2) ^ 2;
MN ^ 2 = x ^ 2 + (1/2 * x – 1/2 – 1) ^ 2 + (2 – 2) ^ 2;
MN ^ 2 = x ^ 2 + (1/2 * x – 3/2) ^ 2;
MN ^ 2 = x ^ 2 + 1/4 * x ^ 2 – 3/2 * x + 9/4;
4MN ^ 2 = 4x ^ 2 + x ^ 2 – 6x + 9;
4MN ^ 2 = 5x ^ 2 – 6x + 9.
2. The smallest value MN (MH) is the distance from the point M to the straight line:

f (x) = 5x ^ 2 – 6x + 9;
D / 4 = 3 ^ 2 – 5 * 9 = 9 – 45 = -36;
xmin = -b / 2a = 6/10 = 3/5;
fmin = -D / 4a = 36/5;
4MH ^ 2 = fmin = 36/5;
MH ^ 2 = 9/5;
MH = 3 / √5.
3. Projection of point M (point H):

x0 = xmin = 3/5 = 0.6;
y0 = 1/2 * x0 – 1/2 = 1/2 * 3/5 – 1/2 = 0.3 – 0.5 = -0.2;
z0 = 2.
H (0.6; -0.2; 2).
Answer: H (0.6; -0.2; 2); MH = 3 / √5.