Sphere Diameter Endpoints Calculation

The problem asks us to find the value of \(u + v + w\) for a sphere given by the equation \(x^2 + y^2 + z^2 + 2ux + 2vy + 2wz - 1 = 0\). We are provided with the coordinates of the two endpoints of a diameter of this sphere: \((1, -1, 2)\) and \((2, 1, -1)\).

Finding the Center of the Sphere

A key property of a sphere is that the midpoint of any diameter is the center of the sphere. The general equation of a sphere is given as \(x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0\). The center of a sphere in this form is located at the coordinates \((-u, -v, -w)\).

Let the two endpoints of the diameter be \(P_1 = (x_1, y_1, z_1) = (1, -1, 2)\) and \(P_2 = (x_2, y_2, z_2) = (2, 1, -1)\). We can find the midpoint \(M\) using the midpoint formula:

\(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)\)

Substituting the coordinates of \(P_1\) and \(P_2\):

\(M = \left( \frac{1 + 2}{2}, \frac{-1 + 1}{2}, \frac{2 + (-1)}{2} \right)\)

\(M = \left( \frac{3}{2}, \frac{0}{2}, \frac{1}{2} \right)\)

\(M = \left( \frac{3}{2}, 0, \frac{1}{2} \right)\)

This midpoint \(M\) represents the center of the sphere.

Relating Center Coordinates to u, v, w

We know the center of the sphere is \((-u, -v, -w)\) and we just calculated it to be \(\left( \frac{3}{2}, 0, \frac{1}{2} \right)\). By equating the corresponding coordinates, we get:

• \(-u = \frac{3}{2}\)
• \(-v = 0\)
• \(-w = \frac{1}{2}\)

Solving for \(u\), \(v\), and \(w\):

• \(u = -\frac{3}{2}\)
• \(v = 0\)
• \(w = -\frac{1}{2}\)

Calculating u + v + w

Finally, we need to calculate the sum \(u + v + w\).

\(u + v + w = \left(-\frac{3}{2}\right) + (0) + \left(-\frac{1}{2}\right)\)

\(u + v + w = -\frac{3}{2} - \frac{1}{2}\)

\(u + v + w = -\frac{3 + 1}{2}\)

\(u + v + w = -\frac{4}{2}\)

\(u + v + w = -2\)

Summary of Results

The calculation shows that the value of \(u + v + w\) is -2.