Show that the sphere through the noncoplanar points 𝐐(x1, y1, z1), 𝐑(x2, y2, z2), 𝐒(x3, y3, z3),

step 1 Hello. So the equation of the sphere in three space is given by C1 times the quantity X squared

Show that the sphere through the noncoplanar points 𝐐(x1,...

Show that the sphere through the noncoplanar points $\mathbf{Q}\left(x_1, y_1, z_1\right)$, $\mathbf{R}\left(x_2, y_2, z_2\right), \mathbf{S}\left(x_3, y_3, z_3\right)$, and $\mathbf{T}\left(x_4, y_4, z_4\right)$ has equation $$ \left|\begin{array}{lllll} x^2+y^2+z^2 & x & y & z & 1 \\ x_1^2+y_1^2+z_1^2 & x_1 & y_1 & z_1 & 1 \\ x_2^2+y_2^2+z_2^2 & x_2 & y_2 & z_2 & 1 \\ x_3^2+y_3^2+z_3^2 & x_3 & y_3 & z_3 & 1 \\ x_4^2+y_4^2+z_4^2 & x_4 & y_4 & z_4 & 1 \end{array}\right|=0 $$

Show that the sphere through the noncoplanar points $\mathbf{Q}\left(x_1, y_1, z_1\right)$, $\mathbf{R}\left(x_2, y_2, z_2\right), \mathbf{S}\left(x_3, y_3, z_3\right)$, and $\mathbf{T}\left(x_4, y_4, z_4\right)$ has equation

$$
\left|\begin{array}{lllll}
x^2+y^2+z^2 & x & y & z & 1 \\
x_1^2+y_1^2+z_1^2 & x_1 & y_1 & z_1 & 1 \\
x_2^2+y_2^2+z_2^2 & x_2 & y_2 & z_2 & 1 \\
x_3^2+y_3^2+z_3^2 & x_3 & y_3 & z_3 & 1 \\
x_4^2+y_4^2+z_4^2 & x_4 & y_4 & z_4 & 1
\end{array}\right|=0
$$

Key Concepts

Sphere Equation in Three Dimensions

In three-dimensional geometry, a sphere is defined as the set of all points that are equidistant from a given center. Its standard equation, (x - h)² + (y - k)² + (z - l)² = R², relates the coordinates of any point on the sphere to the center (h, k, l) and the radius R. This formula serves as the starting point for deriving various algebraic representations of a sphere.

Determinant Method for Spheres Through Points

The determinant method provides an elegant algebraic tool to represent the sphere that passes through a given set of points. By constructing a matrix whose rows combine quadratic and linear terms of the coordinate variables and setting its determinant to zero, one encapsulates the condition that a point lies on the sphere defined by the four given points. This representation simplifies the verification and derivation of the sphere’s equation in a unified form.

Non-Coplanarity and Unique Sphere Determination

The condition that the four points are non-coplanar is critical because it ensures that they do not all lie on a single plane, which in turn guarantees that there is a unique sphere passing through them. If the points were coplanar, the determinant approach would lead to a degenerate case, failing to uniquely specify a sphere. Thus, non-coplanarity underpins the validity and applicability of this method in defining a unique sphere.

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