Let a = 2 i ^ + j ^ + k ^ , and b and c be non-zero vectors such that | a + b + c | = | a + b - c | and b · c = 0 . Consider the following two statements: (A) | a + c | | a | for all R . (B) a and c are always parallel. Then,

Question

Let $\vec{a} = 2 \hat{i} + \hat{j} + \hat{k}$ , and $\vec{b}$ and $\vec{c}$ be two non-zero vectors such that $| \vec{a} + \vec{b} + \vec{c} |$ $=$ $| \vec{a} + \vec{b} - \vec{c} |$ and $\vec{b} \cdot \vec{c} = 0$ .

Consider the following two statements:

(A) $| \vec{a} + λ \vec{c} | \geq | \vec{a} |$ for all $λ \in R$ .

(B) $\vec{a}$ and $\vec{c}$ are always parallel.

Then, [2023]

Smart Achievers · Accepted Answer

(2)

$We have,$ $| \vec{a} + \vec{b} + \vec{c} |$ $\vec{b}$ $| \vec{a} + \vec{b} - \vec{c} |$

$| \vec{a} + \vec{b} + \vec{c} |^{2} = | \vec{a} + \vec{b} - \vec{c} |^{2}$

$| \vec{a} |^{2} + | \vec{b} |^{2} + | \vec{c} |^{2} + 2 \vec{a} \cdot \vec{b} + 2 \vec{b} \cdot \vec{c} + 2 \vec{c} \cdot \vec{a}$

$= | \vec{a} |^{2} + | \vec{b} |^{2} + | \vec{c} |^{2} + 2 \vec{a} \cdot \vec{b} - 2 \vec{b} \cdot \vec{c} - 2 \vec{c} \cdot \vec{a}$

$\Rightarrow 0 + 2 \vec{c} \cdot \vec{a} = - 0 - 2 \vec{c} \cdot \vec{a}$ $[∵ \vec{b} \cdot \vec{c} = 0]$

$\Rightarrow \vec{c} \cdot \vec{a} = 0 \Rightarrow \vec{a} and \vec{c} are perpendicular.$

$Hence (B) is incorrect.$

$Now, | \vec{a} + λ \vec{c} |^{2} \geq | \vec{a} |^{2}$

$\Rightarrow | \vec{a} |^{2} + λ^{2} | \vec{c} |^{2} + 2 λ \vec{a} \cdot \vec{c} \geq | \vec{a} |^{2}$

$\Rightarrow λ^{2} | \vec{c} |^{2} \geq 0 (∵ \vec{a} \cdot \vec{c} = 0)$

$We have \vec{c} \neq 0 \Rightarrow λ^{2} | \vec{c} |^{2} > 0 is true for λ \in R$ , $∴ λ^{2} | \vec{c} |^{2} \geq 0 is also true.$

$Hence (A) is correct.$

Solution

1 both (A) and (B) are correct

2 only (A) is correct

3 only (B) is correct

4 neither (A) nor (B) is correct

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