The length of the shortest distance between the lines r - 3 i - 8 j - 7 k - - 1 - 2 j - k and r - - i - j - k - - 7 i - 6 j - k is

The correct option is D 134√( 116 ) r⃗=3i+a_1↑8j+7k+λ (1 2Jb_1↑+k) r⃗= i a_2↑j k+u(7i b_2↑6j+k) (b⃗_⃗1⃗×b⃗_⃗2⃗).(a⃗_⃗2⃗ a⃗_⃗1⃗)|b⃗_⃗1⃗×b⃗_⃗ ...

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Byju's Answer

Standard XII

Mathematics

Shortest Distance between Two Skew Lines

The length of...

Question

The length of the shortest distance between the lines $r = 3 i + 8 j + 7 k + λ (1 - 2 j + k)$ and $r = - i - j - k + μ (7 i - 6 j + k)$ is

83

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\sqrt{6}

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\sqrt{3}

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\frac{134}{\sqrt{116}}

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Solution

The correct option is D $\frac{134}{\sqrt{116}}$
$\to r = 3 i +_{1} 8 j + 7 k + λ (1 - 2 J_{1} + k)$
$\to r = - i -_{2} j - k + u (7 i -_{2} 6 j + k)$
$\frac{(_{1} \times_{2}) . (_{2} -_{1})}{|_{1} \times_{2} |}$
$b_{1} \times b_{2}$ $⎡ ⎢ ⎣ \begin{matrix} i & j & k 1 & - 2 & 1 7 & - 6 & 1 \end{matrix} ⎤ ⎥ ⎦$
$i (- 2 + 6) - j (1 - 7) + k (- 6 + 14)$
$4 i + 6 j + 8 k$
$_{1} -_{1} = [(3 i + 8 j + 7 k) - (- i - j - k)]$
$= - [4 i + 9 j + 8 k]$
$(b_{1} \times b_{2}) . (_{2} -_{1})$
$[4 i + 6 j + 8 k] . [4 i + 9 j + 8 k]$
$= (16 + 54 + 64) = - 134$
$| b_{1} \times b_{2} | = \sqrt{(4)^{2} + (6)^{2} + (6)^{2}} = | \frac{- 134}{\sqrt{116}} |$
$= d = \frac{134}{\sqrt{116}}$ .

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Similar questions

The shortest distance between the lines $\to r = 3 \to i + 5 \to j + 7 \to k + λ (\to i + 2 \to j + \to k) a n d \to r = - \to i - \to j - \to k + μ (7 \to i - 6 \to j + \to k) i s$

Q. Find the shortest distance between the following pairs of lines whose vector equations are:

(i)

\vec{r} = 3 \hat{i} + 8 \hat{j} + 3 \hat{k} + λ (3 \hat{i} - \hat{j} + \hat{k}) and \vec{r} = - 3 \hat{i} - 7 \hat{j} + 6 \hat{k} + μ (- 3 \hat{i} + 2 \hat{j} + 4 \hat{k})

(ii)

\vec{r} = (3 \hat{i} + 5 \hat{j} + 7 \hat{k}) + λ (\hat{i} - 2 \hat{j} + 7 \hat{k}) and \vec{r} = - \hat{i} - \hat{j} - \hat{k} + μ (7 \hat{i} - 6 \hat{j} + \hat{k})

(iii)

\vec{r} = (\hat{i} + 2 \hat{j} + 3 \hat{k}) + λ (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) and \vec{r} = (2 \hat{i} + 4 \hat{j} + 5 \hat{k}) + μ (3 \hat{i} + 4 \hat{j} + 5 \hat{k})

(iv)

\vec{r} = (1 - t) \hat{i} + (t - 2) \hat{j} + (3 - t) \hat{k} and \vec{r} = (s + 1) \hat{i} + (2 s - 1) \hat{j} - (2 s + 1) \hat{k}

(v)

\vec{r} = (λ - 1) \hat{i} + (λ + 1) \hat{j} - (1 + λ) \hat{k} and \vec{r} = (1 - μ) \hat{i} + (2 μ - 1) \hat{j} + (μ + 2) \hat{k}

(vi)

\vec{r} = (2 \hat{i} - \hat{j} - \hat{k}) + λ (2 \hat{i} - 5 \hat{j} + 2 \hat{k}) and, \vec{r} = (\hat{i} + 2 \hat{j} + \hat{k}) + μ (\hat{i} - \hat{j} + \hat{k})

(vii)

\vec{r} = (\hat{i} + \hat{j}) + λ (2 \hat{i} - \hat{j} + \hat{k}) and, \vec{r} = 2 \hat{i} + \hat{j} - \hat{k} + μ (3 \hat{i} - 5 \hat{j} + 2 \hat{k})

(viii)

\vec{r} = (8 + 3 λ) \hat{i} - (9 + 16 λ) \hat{j} + (10 + 7 λ) \hat{k}

and

\vec{r} = 15 \hat{i} + 29 \hat{j} + 5 \hat{k} + μ (3 \hat{i} + 8 \hat{j} - 5 \hat{k})

[NCERT EXEMPLAR]

Q. Find the shortest distance and the vector equation of the line of shortest distance between the lines given by:

\to r = (3 i + 8 j + 3 k) + λ (3 i - j + k)

\to r = (- 3 i - 7 j + 6 k) + μ (- 3 i + 2 j + 4 k)

Q. Find the shortest distance and the equation of the line corresponding to the shortest distance between the lines.

\to r = (3 i + 8 j + k) + λ (3 i - j + k)

\to r = (- 3 i - 7 j + 6 k) + μ (3 i 2 j + 4 k)

Q. Find the shortest distance between the following pairs of parallel lines whose equations are:
(i)

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

(ii)

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

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The length of the shortest distance between the lines r=3i+8j+7k+λ(1−2j+k) and r=−i−j−k+μ(7i−6j+k) is

The length of the shortest distance between the lines $r = 3 i + 8 j + 7 k + λ (1 - 2 j + k)$ and $r = - i - j - k + μ (7 i - 6 j + k)$ is