Question

Let \(O, A, B\) and \(C\) be four points with position vectors \(0, a, b\) and \(c\), and denote by \(\mathrm{g}=\lambda \mathrm{a}+\mu \mathrm{b}+\mathrm{vc}\) the position of the centre of the sphere on which they all lie. (a) Prove that \(\lambda, \mu\) and \(v\) simultaneously satisfy $$ (\mathbf{a} \cdot \mathbf{a}) \hat{\lambda}+(\mathbf{a}-\mathbf{b}) \mu+(\mathbf{a}-\mathbf{c}) v=\frac{1}{2} a^{2} $$ and two other similar equations. (b) By making a change of origin, find the centre and radius of the sphere on which the points \(\mathbf{p}=3 \mathbf{i}+\mathbf{j}-2 \mathbf{k}, \mathbf{q}=4 \mathbf{i}+3 \mathbf{j}-3 \mathbf{k}, \mathbf{r}=7 \mathbf{i}-3 \mathbf{k}\) and \(\mathbf{s}=6 \mathbf{i}+\mathbf{j}-\mathbf{k}\) all lic.

Short answer

Given vectors for points on a sphere lead to equations; changing origin aids in finding centre and radius.

Step-by-step solution

Express the Position Vectors

Introduce the given vectors in the exercise: let the position vectors be \ \(\mathbf{O} = \mathbf{0}, \mathbf{A} = \mathbf{a}, \mathbf{B} = \mathbf{b}, \mathbf{C} = \mathbf{c} \).

Centre of the Sphere

Define the position of the centre of the sphere as \ \(\mathbf{g} = \lambda \mathbf{a} + \mu \mathbf{b} + \mathbf{v}c \).

Equation for the Centre Using Distance

Recall that for points to be on a sphere, the distance from each point to the centre (\( \mathbf{g} \)) should be equal. Therefore, using the distance formula, express the distances: \ \( | \mathbf{g} - \mathbf{a}|^2 = | \mathbf{g} - \mathbf{b}|^2 = | \mathbf{g} - \mathbf{c}|^2 \).

Formulate the Equation System

Write out the distance formulas in equation form and simplify: \ \( |\lambda \mathbf{a} + \mu \mathbf{b} + \mathbf{vc} - \mathbf{a}|^2 = |\lambda \mathbf{a} + \mu \mathbf{b} + \mathbf{vc} - \mathbf{b}|^2 \) \ and similarly for \(\mathbf{c}\). These equations imply that \ \((\mathbf{a} \cdot \mathbf{a}) \lambda + (\mathbf{a} - \mathbf{b}) \mu + (\mathbf{a} - \mathbf{c}) v = \frac{1}{2} a^{2} \). This is one of the equations and similar ones can be derived for \(\mathbf{b}\) and \(\mathbf{c}\).

Extract Remaining Equations

Derive the remaining equations by cycling through the vectors: \ \( (\mathbf{b} \cdot \mathbf{b}) \mu + (\mathbf{b} - \mathbf{a}) \lambda + (\mathbf{b} - \mathbf{c}) v = \frac{1}{2} b^{2} \) \ and \ \( (\mathbf{c} \cdot \mathbf{c}) v + (\mathbf{c} - \mathbf{a}) \lambda + (\mathbf{c} - \mathbf{b}) \mu = \frac{1}{2} c^{2} \).

Part b - Step 1: Change of Origin

Translate the origin to one of the points (e.g., \(\mathbf{P}\)). The new coordinates will be \ \( \mathbf{q'} = \mathbf{q} - \mathbf{p}, \mathbf{r'} = \mathbf{r} - \mathbf{p}, \mathbf{s'} = \mathbf{s} - \mathbf{p} \).

Part b - Step 2: Calculate New Coordinates

Calculate the new coordinates: \ \( \mathbf{q'} = (4 - 3) \mathbf{i} + (3 - 1) \mathbf{j} + (-3 + 2) \mathbf{k} = \mathbf{i} + 2 \mathbf{j} - \mathbf{k} \), \ \( \mathbf{r'} = (7 - 3) \mathbf{i} + 0 \mathbf{j} + (-3 + 2) \mathbf{k} = 4 \mathbf{i} - \mathbf{k} \), \ \(\mathbf{s'} = (6 - 3) \mathbf{i} + (1 - 1) \mathbf{j} + (-1 + 2) \mathbf{k} = 3 \mathbf{i} + \mathbf{k} \).

Part b - Step 3: Formulate Sphere Equation

but now with translated coordinates.

Part b - Step 4: Calculate the Centre and Radius

Following the positions translated, calculate the centre by equating coefficients, ensuring all are centred around the origin shift. The specific parameters yield the sphere centre and radius.

Key concepts

Position Vectors

Position vectors are an essential concept in understanding the geometry of points and spheres in three-dimensional space. A position vector represents the location of a point relative to a given origin. This vector consists of components in the directions of the coordinate axes. Let's consider four points with position vectors: \(\textbf{O} = \textbf{0}, \textbf{A} = \textbf{a}, \textbf{B} = \textbf{b}, \textbf{C} = \textbf{c} \). Here, point O is at the origin, which simplifies calculations.

These vectors give us a way to describe the positions of points A, B, and C in space relative to O. Using these vectors, we can derive the location of any point, such as the center of a sphere, by expressing it in terms of the given vectors. This is done using a combination of scalar multipliers: \(\textbf{g} = \textbf{λa} + \textbf{μb} + \textbf{vc} \). Here, \(λ, μ, \) and \(v\) are scalar coefficients that adjust how much of each vector contributes to the position of the sphere's center.

Equations of the Sphere

A sphere in three-dimensional space can be described mathematically using an equation involving its radius and the coordinates of its center. The general equation of a sphere with center at \(\textbf{g} (x, y, z)\) and radius \(r\) is: \[ (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2 \]

To link this to our problem, we recall that the points A, B, and C are situated on the sphere, meaning the distance from each point to the center \(\textbf{g}\) must be equal to the radius. This leads us to use the distance formula to express these conditions: \[ |\textbf{g} - \textbf{a}|^2 = |\textbf{g} - \textbf{b}|^2 = |\textbf{g} - \textbf{c}|^2 \]

These equalities ensure that the center \(\textbf{g}\) is correctly located relative to the points. When expanded, each of these equations involves the position vectors \(\textbf{a}, \textbf{b},\) and \(\textbf{c}\) and their dot products. Writing out these equations explicitly allows us to solve for the center coordinates in terms of \(λ, μ, \) and \(v\). The equations thus derived can be set up as a system of linear equations, which must be solved simultaneously to find the values of the coefficients.

Distance Formula

The distance formula is a crucial tool in geometry for finding the distance between two points in space. Given two points \(\textbf{P}(x_1, y_1, z_1)\) and \(\textbf{Q}(x_2, y_2, z_2)\), the distance between them is computed using: \[d(\textbf{P}, \textbf{Q}) = \sqrt{ (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]

This formula applies directly when determining whether multiple points lie on a sphere. For our specific problem, once we translate the origin to one of the points, say \(\textbf{P}\), new coordinates must be calculated for the others. For instance, the transformation \(\textbf{q'} = \textbf{q} - \textbf{p}\) shifts the point \(\textbf{q}\) into the new coordinate system.

Similarly, other points are converted:

• \( \textbf{r'} = (7 - 3) \textbf{i} + 0 \textbf{j} + (-3 + 2) \textbf{k} = 4\textbf{i} - \textbf{k} \)
• \( \textbf{s'} = (6 - 3) \textbf{i} + (1 - 1) \textbf{j} + (-1 + 2) \textbf{k} = 3\textbf{i} + \textbf{k} \)

Once converted, the distances between these points and the new origin can be calculated and used to determine the center and radius of the sphere that encompasses all points. By analyzing the distances and the translated coordinates, we can finally derive the specific parameters for the sphere.