The general value of x^2+y^2+z^2 where x, y, z represent a non-zero solution of x+2 y-3 z=0,2 x-

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The general value of x^2+y^2+z^2 where x, y, z represent a...

The general value of $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}$ where $\mathrm{x}, \mathrm{y}, \mathrm{z}$ represent a non-zero solution of $\mathrm{x}+2 \mathrm{y}-3 \mathrm{z}=0,2 \mathrm{x}-3 \mathrm{y}+\mathrm{z}=0,3 \mathrm{x}-\mathrm{y}$ $-2 \mathrm{z}=0$ is (where $\lambda$ is an arbitrary constant) (a) 3 (b) $3 \lambda$ (c) $9 \lambda^{2}$ (d) $3 \lambda^{2}$

The general value of $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}$ where $\mathrm{x}, \mathrm{y}, \mathrm{z}$ represent a non-zero solution of $\mathrm{x}+2 \mathrm{y}-3 \mathrm{z}=0,2 \mathrm{x}-3 \mathrm{y}+\mathrm{z}=0,3 \mathrm{x}-\mathrm{y}$
$-2 \mathrm{z}=0$ is (where $\lambda$ is an arbitrary constant)
(a) 3
(b) $3 \lambda$
(c) $9 \lambda^{2}$
(d) $3 \lambda^{2}$

Key Concepts

Homogeneous Systems of Linear Equations

A homogeneous system of linear equations is one in which all constant terms are zero. Such systems always have at least the trivial solution, but non-trivial solutions exist if the system's coefficient matrix is singular. In this context, the non-zero solution is represented in terms of an arbitrary constant, showing that the system has infinitely many solutions, each being a scalar multiple of a basis vector.

Parametric Representation of Solutions

When a system of equations has free parameters, the solutions can be expressed in a parametric form. This means that the variables are written in terms of an arbitrary constant (or constants), effectively capturing all possible solutions. In the problem, the use of an arbitrary constant (commonly denoted by lambda) represents how each component of the solution vector is scaled uniformly, reflecting the infinite families of solutions inherent in homogeneous systems.

Vector Norm (Magnitude)

The norm of a vector, often computed as the square root of the sum of the squares of its components, measures the length or size of the vector. In problems involving parametric solutions, the norm may depend on the arbitrary constant introduced in the parameterization. Here, computing the square of the norm (x² + y² + z²) leads to an expression that is proportional to the square of the arbitrary constant, a typical result when all components of the vector are scaled by the same parameter.

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