Two different vectors having same magnitude :

ˆ i + 2 ˆ j + 3 ˆ k, 3 ˆ i + 2 ˆ j + ˆ k

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ˆ i + ˆ j - ˆ k, 2 ˆ i - ˆ j - ˆ k

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ˆ i - 2 ˆ j + 2 ˆ k, ˆ i + ˆ j + ˆ k

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None

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Solution

The correct option is A $ˆ i + 2 ˆ j + 3 ˆ k, 3 ˆ i + 2 ˆ j + ˆ k$
Option : A
$(^i + 2^j + 3^k)$ $(3^i + 2^j +^k)$
$\Rightarrow ∣ ∣^i + 2^j + 3^k ∣ ∣ = \sqrt{1 + 4 + 9}$
$= \sqrt{14}$
$\Rightarrow ∣ ∣ 3^i + 2^j +^k ∣ ∣ = \sqrt{1 + 4 + 9}$
$= \sqrt{14}$ Correct
Option : B
$(^i +^j -^k)$ $(2^i -^j -^k)$
$\Rightarrow ∣ ∣^i +^j -^k ∣ ∣ = \sqrt{1 + 1 + 1}$
$= \sqrt{3}$
$\Rightarrow ∣ ∣ 2^i -^j -^k ∣ ∣ = \sqrt{4 + 1 + 1}$
$= \sqrt{6}$ Incorrect
Option : C
$(^i - 2^j + 2^k)$ $(^i +^j +^k)$
$\Rightarrow ∣ ∣^i - 2^j + 2^k ∣ ∣ = \sqrt{1 + 4 + 4}$
$= 3$
$\Rightarrow ∣ ∣^i +^j +^k ∣ ∣ = \sqrt{1 + 1 + 1}$
$= \sqrt{3}$
Hence the answer is $(^i + 2^j + 3^k), (3^i + 2^j +^k)$

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[(3 i - 2 j - 2 k) \times (i - k)] \times [(i + j + k) \times (i - 2 j + 3 k)]

¯ ¯¯¯¯¯¯ ¯ O A = i + j + k, ¯ ¯¯¯¯¯¯ ¯ A B = 3 i - 2 j + k, ¯ ¯¯¯¯¯¯ ¯ B C = i + 2 j - 2 k

¯ ¯¯¯¯¯¯¯ ¯ C D = 2 i + j + 3 k

¯ ¯¯¯¯¯¯¯ ¯ O D

\to O A = i + j + k, \to A B = 3 i - 2 j + k, \to B C = i + 2 j - 2 k, \to C D = 2 i + j + 3 k

2 \hat{i} + \hat{j} - \hat{k}, 3 \hat{i} - 2 \hat{j} + \hat{k} and \hat{i} + 4 \hat{j} - 3 \hat{k}

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\vec{a} = \hat{i} - \hat{j} + \hat{k}, \vec{b} = 2 \hat{i} + \hat{j} - \hat{k}, \vec{c} = λ \hat{i} - \hat{j} + λ \hat{k}

\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} - 3 \hat{k}, \vec{c} = λ \hat{i} + λ \hat{j} + 5 \hat{k}

\vec{a} = \hat{i} + 2 \hat{j} - 3 \hat{k}, \vec{b} = 3 \hat{i} + λ \hat{j} + \hat{k}, \vec{c} = \hat{i} + 2 \hat{j} + 2 \hat{k}

\vec{a} = \hat{i} + 3 \hat{j}, \vec{b} = 5 \hat{k}, \vec{c} = λ \hat{i} - \hat{j}

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