JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Vector Algebra

For any vector

a = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}

, with

∣ a_{i} ∣ < 1, i = 1, 2, 3

, consider the following statements:
(A):

max {∣ a_{1} ∣, ∣ a_{2} ∣, ∣ a_{3} ∣} \leq ∣ a ∣

(B):

∣ a ∣ \leq 3 max {∣ a_{1} ∣, ∣ a_{2} ∣, ∣ a_{3} ∣}

Answer: (D)

Solution

Let

∣ a_{3} ∣ = max {∣ a_{1} ∣, ∣ a_{2} ∣, ∣ a_{3} ∣}

. For (A):

∣ a ∣^{2} = ∣ a_{1} ∣^{2} + ∣ a_{2} ∣^{2} + ∣ a_{3} ∣^{2} \geq ∣ a_{3} ∣^{2}

, so

∣ a ∣ \geq ∣ a_{3} ∣

. Statement (A) is true. For (B):

∣ a ∣^{2} \leq ∣ a_{3} ∣^{2} + ∣ a_{3} ∣^{2} + ∣ a_{3} ∣^{2} = 3∣ a_{3} ∣^{2}

, so

∣ a ∣ \leq 3 ∣ a_{3} ∣ < 3∣ a_{3} ∣

. Statement (B) is true. Both statements are true.

JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Single correct option