Part BCSIR NET December 2024isotropic-vectors-exist-in-an-indefinite-form

Isotropic vectors exist in an indefinite form

Consider the real matrix A = [[0, 0, 1], [0, 1, 0], [1, 0, 0]]. Define $B : \mathbb{R}^{3} \times \mathbb{R}^{3} \to \mathbb{R}$ by B(v, w) = vᵗAw. Which of the following statements is true?

A.B(v, v) = 0 if and only if v = 0.
B.For every $\lambda \in \mathbb{R}$ , there exists v such that $B(v, v) = \lambda$ .✓ Correct
C.There exists v ≠ 0 such that B(v, w) = 0 for all $w \in \mathbb{R}^{3}$ .
D.If B(v, w) = 0 then either v = 0 or w = 0.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Isotropic vectors exist in an indefinite form

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