Mathematics Question from JEE Main 2023 · 31 January, Shift 1

Let $\mathbf{a} = 2\hat{i} + \hat{j} + \hat{k}$, and $\mathbf{b}$ and $\mathbf{c}$ be two nonzero vectors such that

$$\left| \mathbf{a} + \mathbf{b} + \mathbf{c} \right| = \left| \mathbf{a} + \mathbf{b} - \mathbf{c} \right| \quad \text{and} \quad \mathbf{b} \cdot \mathbf{c} = 0.$$

Consider the following two statements: (A) The magnitude of the vector $\mathbf{a}$ plus $\lambda$ times the vector $\mathbf{c}$ is greater than or equal to the magnitude of vector $\mathbf{a}$, for all real numbers $\lambda$. (B) $\mathbf{a}$ and $\mathbf{c}$ are always parallel.