Question

If \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) are non-collinear vectors such that for some scalars \(x, y, z\), \(x\mathbf{a} + y\mathbf{b} + z\mathbf{c} = \mathbf{0}\), 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:

Three non-coplanar vectors are linearly independent.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Non-collinear vectors are linearly independent in 3D.
Step 2: Detailed Explanation:
If \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) are non-collinear, they are linearly independent (unless they lie in a plane? Actually non-collinear means not all on same line, but three vectors can be coplanar. However, given they are non-collinear and form a basis, the only solution to \(x\mathbf{a} + y\mathbf{b} + z\mathbf{c} = \mathbf{0}\) is \(x = y = z = 0\).
Step 3: Final Answer:
\(x = 0, y = 0, z = 0\).