Find the shortest distance between the skew lines $r = (6 i + 2 j + 2 k) + t (i - 2 j + 2 k)$ and $r = (- 4 i - k) + s (3 i - 2 j - 2 k)$ .

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Solution

$\begin{matrix} \to r = (6 ˆ i + 2 ˆ j + 2 ˆ k) + t (ˆ i - 2 ˆ j + 2 ˆ k) r = (- 4 ˆ i - ˆ k) + s (3 ˆ i - 2 ˆ j - 2 ˆ k) S h o r t e s t d i s tan c e d = \frac{∣ ∣ (10 ˆ i + 2 ˆ j + 3 ˆ k) . (ˆ i - 2 ˆ j + 2 ˆ k) \times (3 ˆ i - 2 ˆ j - 2 ˆ k) ∣ ∣}{∣ ∣ (ˆ i - 2 ˆ j + 2 ˆ k) \times (3 ˆ i - 2 ˆ j - 2 ˆ k) ∣ ∣} ∣ ∣ ∣ ∣ ∣ \begin{matrix} ˆ i & ˆ j & ˆ k 1 & - 2 & 2 3 & - 2 & - 2 \end{matrix} ∣ ∣ ∣ ∣ ∣ = ˆ i (4 + 4) - ˆ j (- 2 - 6) + ˆ k (- 2 + 6) = 8 ˆ i + 8 ˆ j + 4 ˆ k = \frac{∣ ∣ (10 ˆ i + 2 ˆ j + 3 ˆ k) . (8 ˆ i + 8 ˆ j + 4 ˆ k) ∣ ∣}{\sqrt{64 + 64 + 16}} S h o r t e s t d i s tan e = \frac{80 + 16 + 12}{\sqrt{144}} = \frac{108}{12} = 9 \end{matrix}$

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Similar questions

r = (6 i + 2 j + 2 k) + t (i - 2 j + 2 k)

F = (- 4 i - k) + s (3 i - 2 j - 2 k)

\to r = (6 ˆ i + 2 ˆ j + 2 ˆ k) + λ (ˆ i - 2 ˆ j + 2 ˆ k)

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

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

\frac{x + 1}{7} = \frac{y + 1}{- 6} = \frac{z + 1}{1} and \frac{x - 3}{1} = \frac{y - 5}{- 2} = \frac{z - 7}{1}

\vec{r} = \hat{i} + 2 \hat{j} + 3 \hat{k} + λ (\hat{i} - 3 \hat{j} + 2 \hat{k}) and \vec{r} = 4 \hat{i} + 5 \hat{j} + 6 \hat{k} + μ (2 \hat{i} + 3 \hat{j} + \hat{k})

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

¯ ¯ ¯ r = 4 ¯ i - ¯ j + λ (¯ i + 2 ¯ j - 5 ¯ ¯ ¯ k)

¯ ¯ ¯ r = ¯ i - ¯ j + 2 ¯ ¯ ¯ k + μ (¯ i + 2 ¯ j - 5 ¯ ¯ ¯ k)

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

\vec{r} = (3 \hat{i} + 2 \hat{j} - 4 \hat{k}) + λ (\hat{i} + 2 \hat{j} + 2 \hat{k}) and \vec{r} = (5 \hat{j} - 2 \hat{k}) + μ (3 \hat{i} + 2 \hat{j} + 6 \hat{k})

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