Let a⇀= i⇀+xj⇀+k⇀, b⇀= i⇀+j⇀+k⇀ and a⇀+b⇀=a⇀+b⇀. Then the possible values of x are

a⇀+b⇀=a⇀+b⇀⇔ a⇀, b⇀ are like vectors.⇔a⇀=λb⇀ for some λ∈R⇔i⇀+xj⇀+k⇀=λi⇀+j⇀+k⇀⇔x=λ=1

Let a⇀= i⇀+xj⇀+k⇀, b⇀= i⇀+j⇀+k⇀ and a⇀+b⇀=a⇀+b⇀. Then the possible values of x are

a⇀+b⇀=a⇀+b⇀⇔ a⇀, b⇀ are like vectors.⇔a⇀=λb⇀ for some λ∈R⇔i⇀+xj⇀+k⇀=λi⇀+j⇀+k⇀⇔x=λ=1

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Let $\overset{⇀}{a}$ = $\overset{⇀}{i}$ $+$ x $\overset{⇀}{j}$ $+ \overset{⇀}{k}$ , $\overset{⇀}{b} = \overset{⇀}{i} + \overset{⇀}{j} + \overset{⇀}{k}$ and $|\overset{⇀}{a} + \overset{⇀}{b}| = |\overset{⇀}{a}| + |\overset{⇀}{b}|$ . Then the possible values of x are

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x $=$ 1

x $= - 1$

all real values of x

no real values of x

answer is A.

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Detailed Solution

$|\overset{⇀}{a} + \overset{⇀}{b}| = |\overset{⇀}{a}| + |\overset{⇀}{b}|$

$\Leftrightarrow \overset{⇀}{a}, \overset{⇀}{b}$ are like vectors.

$\Leftrightarrow$ $\overset{⇀}{a} = λ \overset{⇀}{b}$ for some $λ \in$ R

$\Leftrightarrow$ $\overset{⇀}{i} + x \overset{⇀}{j} + \overset{⇀}{k} = λ (\overset{⇀}{i} + \overset{⇀}{j} + \overset{⇀}{k})$

$\Leftrightarrow x = λ = 1$

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Let a⇀= i⇀+xj⇀+k⇀, b⇀= i⇀+j⇀+k⇀ and a⇀+b⇀=a⇀+b⇀. Then the possible values of x are

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Let $\overset{⇀}{a}$ = $\overset{⇀}{i}$ $+$ x $\overset{⇀}{j}$ $+ \overset{⇀}{k}$ , $\overset{⇀}{b} = \overset{⇀}{i} + \overset{⇀}{j} + \overset{⇀}{k}$ and $|\overset{⇀}{a} + \overset{⇀}{b}| = |\overset{⇀}{a}| + |\overset{⇀}{b}|$ . Then the possible values of x are