Find the line through 2 - - 1-3 and perpendicular to each of the lines r - i - j - k - - 2 i - 2 j - k and r - 2 i - j - 3 k - - i - 2 j - 2 k

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Question

Find the line through $(2, - 1, 3)$ and perpendicular to each of the lines $r = i + j - k + λ (2 i - 2 j + k)$ and $r = 2 i - j - 3 k + μ (i + 2 j + 2 k)$

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Solution

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(3, - 1, 2)

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

r = (i + j - k) + λ (2 i - 2 j + k)

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

\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})

\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})

\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})

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

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

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