Question

If \( a, b \), and \( c \) are non-collinear vectors such that for some scalars \( x, y, z \), the equation \[ x a + y b + z c = 0, \quad x a + x b + x c = 0, \] then

• A. \( x = 0, y = 0, z = 0 \)
• B. \( x \neq 0, y \neq 0, z = 0 \)
• C. \( x = 0, y \neq 0, z \neq 0 \)
• D. \( x \neq 0, y \neq 0, z \neq 0 \)

Hint:

When dealing with non-collinear vectors in a system of equations, if the vector sum is zero, all coefficients must be zero.

Answer 1

The Correct Option is A

Step 1: Understand the problem.
We are given two vector equations: \[ x a + y b + z c = 0 \quad \text{and} \quad x a + x b + x c = 0 \] We are tasked with finding the values of \( x \), \( y \), and \( z \) that satisfy these conditions.

Step 2: Analyze the second equation.
The second equation can be simplified by factoring out \( x \): \[ x (a + b + c) = 0 \] Since \( a \), \( b \), and \( c \) are non-collinear, \( a + b + c \neq 0 \). Therefore, we must have: \[ x = 0 \]

Step 3: Substitute \( x = 0 \) into the first equation.
Substituting \( x = 0 \) into the first equation, we get: \[ 0 \cdot a + y b + z c = 0 \] This simplifies to: \[ y b + z c = 0 \] Since \( b \) and \( c \) are non-collinear, the only solution to this equation is: \[ y = 0 \quad \text{and} \quad z = 0 \]

Step 4: Conclusion.
Thus, \( x = 0 \), \( y = 0 \), and \( z = 0 \), which corresponds to option (A).