Let \(\vec{a}=2 \hat{i}+\hat{j}+\hat{k}\), and \(\vec{b}\) and \(\vec{c}\) be two nonzero vectors such that \(|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|\) and \(\overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{c}}=0\). Consider the following two statement: (A) \(|\overrightarrow{\mathrm{a}}+\lambda \overrightarrow{\mathrm{c}}| \geq|\overrightarrow{\mathrm{a}}|\) for all \(\lambda \in \mathbb{R}\). (B) \(\vec{a}\) and \(\vec{c}\) are always parallel

(B) neither (A) nor (B) is correct

(D) both (A) and (B) are correct.

JEE Main · 2023 · 31 Jan 2023 · Shift 1 · Vector Algebra

Let

a = 2 \hat{i} + \hat{j} + \hat{k}

, and

b

and

c

be two nonzero vectors such that

∣ a + b + c ∣ = ∣ a + b - c ∣

and

b . c = 0

. Consider the following two statement:
(A)

∣ a + λ c ∣ \geq ∣ a ∣

for all

λ \in R

.
(B)

a

and

c

are always parallel

Answer: (C)

Expanding

∣ a + b + c ∣^{2} = ∣ a + b - c ∣^{2}

leaves

4 (a + b) \cdot c = 0

; since

b \cdot c = 0

is given, this forces

a \cdot c = 0

, so

a

and

c

are perpendicular, not parallel, and statement (B) fails. For statement (A),

∣ a + λ c ∣^{2} = ∣ a ∣^{2} + λ^{2} ∣ c ∣^{2} + 2 λ (a \cdot c) = ∣ a ∣^{2} + λ^{2} ∣ c ∣^{2} \geq ∣ a ∣^{2}

for every real

λ

, so (A) holds. Only statement (A) is correct.

JEE Main · 2023 · 31 Jan 2023 · Shift 1 · Single correct option