The line is parallel to the vector and passes through the point (7, 6, 2) and the line is parallel to the vector and passes through the point (5,3, 4). The shortest distance between the lines and is : [2025]

Question

The line $L_{1}$ is parallel to the vector $\vec{a} = - 3 \hat{i} + 2 \hat{j} + 4 \hat{k}$ and passes through the point (7, 6, 2) and the line $L_{2}$ is parallel to the vector $\vec{b} = 2 \hat{i} + \hat{j} + 3 \hat{k}$ and passes through the point (5,3, 4). The shortest distance between the lines $L_{1}$ and $L_{2}$ is : [2025]

Smart Achievers · Accepted Answer

(1)

We have, $L_{1} : 7 \hat{i} + 6 \hat{j} + 2 \hat{k} + λ (- 3 \hat{i} + 2 \hat{j} + 4 \hat{k})$

$L_{2} : 5 \hat{i} + 3 \hat{j} + 4 \hat{k} + μ (2 \hat{i} + \hat{j} + 3 \hat{k})$

Here, ${\vec{a}}_{1} = 7 \hat{i} + 6 \hat{j} + 2 \hat{k}, {\vec{a}}_{2} = 5 \hat{i} + 3 \hat{j} + 4 \hat{k}$

${\vec{b}}_{1} = - 3 \hat{i} + 2 \hat{j} + 4 \hat{k} and {\vec{b}}_{2} = 2 \hat{i} + \hat{j} + 3 \hat{k}$

Now, ${\vec{b}}_{1} \times {\vec{b}}_{2} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - 3 & 2 & 4 \\ 2 & 1 & 3 \end{matrix} |$

$= \hat{i} (6 - 4) - \hat{j} (- 9 - 8) + \hat{k} (- 3 - 4)$

$= 2 \hat{i} + 17 \hat{j} - 7 \hat{k}$

Also, ${\vec{a}}_{2} - {\vec{a}}_{1} = - 2 \hat{i} - 3 \hat{j} + 2 \hat{k}$

Shortest distance between $L_{1}$ and $L_{2}$

$= \frac{| ({\vec{a}}_{2} - {\vec{a}}_{1}) \cdot ({\vec{b}}_{1} \times {\vec{b}}_{2}) |}{| {\vec{b}}_{1} \times {\vec{b}}_{2} |} = \frac{| - 4 - 51 - 14 |}{\sqrt{4 + 289 + 49}}$

$= \frac{69}{\sqrt{342}} = \frac{69}{3 \sqrt{38}} = \frac{23}{\sqrt{38}}$

Solution

1 $\frac{23}{\sqrt{38}}$

2 $\frac{21}{\sqrt{57}}$

3 $\frac{23}{\sqrt{57}}$

4 $\frac{21}{\sqrt{38}}$

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Solution

Q. The line L1 is parallel to the vector a→=–3i^+2j^+4k^ and passes through the point (7, 6, 2) and the line L2 is parallel to the vector b→=2i^+j^+3k^ and passes through the point (5,3, 4). The shortest distance between the lines L1 and L2 is : [2025]

1 2338

2 2157

3 2357

4 2138

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1 $\frac{23}{\sqrt{38}}$

2 $\frac{21}{\sqrt{57}}$

3 $\frac{23}{\sqrt{57}}$

4 $\frac{21}{\sqrt{38}}$