Which of the following is the direction vector for the shortest distance between the lines L1 and L2, whose vector equations are $_{1} = 2 ˆ i + 3 ˆ j + λ (5 ˆ i + 3 ˆ j - 3 ˆ k)$ and $_{2} = ˆ i + 4 ˆ j + λ (3 ˆ i + 2 ˆ j + ˆ k) .$

-^i +^j

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- 2^i -^j + 4^k

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9^i - 14^j +^k

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- 9^i - 14^j

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Solution

The correct option is C $9^i - 14^j +^k$
Shortest distance between two vectors lies in the direction of $_{1} \times_{2}$
Here, $_{1} = 5 ˆ i + 3 ˆ j - 3 ˆ k$ and
$_{2} = 3 ˆ i + 2 ˆ j + ˆ k$
Taking their cross product
$_{1} \times_{2} = (5 ˆ i + 3 ˆ j - 3 ˆ k) \times (3 ˆ i + 2 ˆ j + ˆ k) = 9 ˆ i - 14 ˆ j + ˆ k$

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Similar questions

_{1} = 2 ˆ i + 3 ˆ j + λ (5 ˆ i + 3 ˆ j - 3 ˆ k)

_{2} = ˆ i + 4 ˆ j + λ (3 ˆ i + 2 ˆ j + ˆ k) .

\vec{r} = (- 2 \hat{i} - 3 \hat{j} + 4 \hat{k}) + λ (3 \hat{i} - 2 \hat{j} - \hat{k})

\hat{i} + 2 \hat{j} + 3 \hat{k}

\vec{r} \cdot (\hat{i} + 3 \hat{k}) = 10

\vec{r} \cdot (\hat{i} - 3 \hat{k}) = 10

\vec{r} \cdot (3 \hat{i} + \hat{k}) = 10

¯ ¯ ¯ r = (2 ¯ i - 3 ¯ j + ¯ ¯ ¯ k) + λ (¯ i + 4 ¯ j + 3 ¯ ¯ ¯ k)

¯ ¯ ¯ r = (¯ i - ¯ j + 2 ¯ ¯ ¯ k) + μ (¯ i + 2 ¯ j - 3 ¯ ¯ ¯ k)

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

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

- 2 \hat{i} + 3 \hat{j}, \hat{i} + 2 \hat{j} + 3 \hat{k} and 7 \hat{i} - \hat{k}

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