$\overrightarrow{\mathrm{A}}$ is a vector quantity such that $|\overrightarrow{\mathrm{A}}|=$ nonzero constant. Which of the following expressions is true for $\overrightarrow{\mathrm{A}}$ ?

(C) $\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{A}}=0$

(A) $\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{A}}=0$

(B) $\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{A}}<0$

(D) $\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{A}}>0$

JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Vector

A

is a vector quantity such that

∣ A ∣ =

nonzero constant. Which of the following expressions is true for

A

(A) $A \cdot A = 0$
(B) $A \times A < 0$
(C) $A \times A = 0$ ✓ Correct
(D) $A \times A > 0$

Answer: (C)

Solution

The cross product of any vector with itself is always zero because the angle between a vector and itself is 0°:

A \times A = ∣ A ∣∣ A ∣ sin (0°) \overset{n}{^} = 0

Note that the dot product

A \cdot A = ∣ A ∣^{2}

, which is nonzero, so option (A) is incorrect. Cross products yield vectors (or zero), not negative scalars, so options (B) and (D) are meaningless.

Answer: (C)

JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Single correct option

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