Final answer:
To find the shortest distance from point P to line PQ, we need to find the equation of the line P1P2 and calculate the distance between point P and a point Q on the line. The shortest distance is the magnitude of the vector connecting them. The shortest distance between point P and line P1P2 is sqrt(10).
Step-by-step explanation:
To find the shortest distance from point P to the line PQ, we can use the formula for finding the distance between a point and a line in 3D space. First, we need to find the equation of the line P1P2. Using the two given points, we can find the direction vector of the line as (3-0,0-0,2-1) = (3,0,1). The equation of the line can be written as P = P1 + t(P2-P1), where t is a parameter. Using the point-slope form, we can write the equation of the line as x = 0 + 3t, y = 0 + 0t, z = 1 + t.
Next, we need to find the vector connecting point P to a point Q on the line P1P2. We can represent this vector as PQ = . To find the point Q, we need to substitute the x, y, and z values of P into the equation of the line. Substituting -3 for x, 1 for y, and 2 for z, we get -3 = 3t, 1 = 0, and 2 = 1 + t. Solving these equations, we find t = -1, x = -3, y = 0, and z = 1.
Therefore, the shortest distance from point P to line PQ is the magnitude of vector PQ, which can be calculated using the formula sqrt(x^2 + y^2 + z^2). Substituting the values, we get sqrt((-3)^2 + 0^2 + (1-2)^2) = sqrt(9 + 0 + 1) = sqrt(10). So, the shortest distance between point P and line P1P2 is sqrt(10).