The adjacent sides of a parallelogram are $2 i + 4 ¯ j - 5 ¯ ¯ ¯ k$ and $i + 2 ¯ j + 3 ¯ ¯ ¯ k$ then the unitvector parallel to a diagonal is

\frac{- i + 2 j + 8 k}{\sqrt{69}}

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\frac{3 i + 6 ¯ j - 2 ¯ ¯ ¯ k}{7}

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\frac{- i + 2 ¯ j - 8 ¯ ¯ ¯ k}{\sqrt{69}}

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\frac{- i - 2 ¯ j + 8 ¯ ¯ ¯ k}{\sqrt{69}}

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Solution

The correct option is C $\frac{3 i + 6 ¯ j - 2 ¯ ¯ ¯ k}{7}$
Sides of parallelogram $\to a = 2 i + 4 j - 5 k \to b = i + 2 j + 3 k$
Diagonal of parallelogram $= \to a + \to b$
Unit vector of diagonal $= \frac{| \to a + \to b |}{\sqrt{(a + b)^{2}}} = \frac{3 i + 6 j - 2 k}{\sqrt{9 + 36 + 4}} = \frac{3 i + 6 j - 2 k}{7}$

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Similar questions

\vec{a} = \hat{i} + 2 \hat{j} - \hat{k}, \vec{b} = 3 \hat{i} + 2 \hat{j} + 7 \hat{k}, \vec{c} = 5 \hat{i} + 6 \hat{j} + 5 \hat{k}

\vec{a} = - 4 \hat{i} - 6 \hat{j} - 2 \hat{k}, \vec{b} = - \hat{i} + 4 \hat{j} + 3 \hat{k}, \vec{c} = - 8 \hat{i} - \hat{j} + 3 \hat{k}

\hat{a} = \hat{i} - 2 \hat{j} + 3 \hat{k}, \hat{b} = - 2 \hat{i} + 3 \hat{j} - 4 \hat{k}, \hat{c} = \hat{i} - 3 \hat{j} + 5 \hat{k}

\to a = 2 i + 4 j - 5 k

\to b = i + 2 j + 3 k

2 i + 4 j - 5 k

i + 2 j + 3 k .

A B C D

2 ˆ i + 4 ˆ j - 5 ˆ k

ˆ i + 2 ˆ j + 3 ˆ k

| - - \to A C \times - - \to B D |

2 \hat{i} - 4 \hat{j} + 5 \hat{k} and \hat{i} - 2 \hat{j} - 3 \hat{k} .

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