Find the shortest distance and the equation of the line corresponding to the shortest distance between the lines.
$\to r = (3 i + 8 j + k) + λ (3 i - j + k)$
$\to r = (- 3 i - 7 j + 6 k) + μ (3 i 2 j + 4 k)$

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Solution

$\begin{matrix} \frac{x - 3}{3} = \frac{y - B}{- 1} = \frac{z - 3}{1} = γ_{1} p (3 r, + 3, B - r_{1}, r_{1} + 3) l_{2} : \frac{x + 3}{- 3} = \frac{y + 7}{2} = \frac{z - 6}{1} = γ_{2} Q = (- 3 - 3 r_{2}, - 7 + 2 r_{2}, 6 + 4 r_{2}) Q P : (3 r_{1} + 3 r_{2} + 6, - r - 2 r_{2} + 15, r_{1} - 4 r_{2} - 3) S o, 3 (3 r_{1} + 3 r_{2} + 6) - (- r_{1} - 2 r_{2} + 15) + (r_{1} - 4 r_{2} - 3) = 0 S i m i l a r l y, 7 r_{1} + 11 r_{2} = 0...... (2) f r o m (1) a n d (2) r_{1} = r_{2} = 0 ∴ P Q = (- 6, - 15, 3) S h o r t e s t d i s tan c e = \sqrt{{(- 3 - 3)}^{2} + {(- 7 - Q)}^{2} + {(6 - 3)}^{2}} = 3 \sqrt{30} d i r e c t i o n r a t i o o f l i n e (2, 5, - 1) S o, e q u a t i o n o f l i n e, \frac{x - 3}{2} = \frac{y - 8}{5} = \frac{z - 3}{- 1} \end{matrix}$

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Similar questions

\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)

r = 3 i + 8 j + 7 k + λ (1 - 2 j + k)

r = - i - j - k + μ (7 i - 6 j + k)

\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})

\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})

\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})

\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}

\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}

\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})

\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})

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

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

\to r = (4, - 1, 0) + k (1, 2, 3), k \in R

\to r = (1, - 1, 2) + k (2, 4, - 5), k \in R

\to r = (4 i - j) + λ (i + 2 j - 3 k)

\to r = (i - j + 2 k) + μ (2 i + 4 j - 5 k)

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