Find the vector equation of a line passing through the point 2-3-2 and parallel to the line r -2 i -3 j - 2 i -3 j -6 k - Also find the distance between these lines-

Vector equation of a line passing through (2, 3, 2) and parallel to the line nbsp;r #8594;= 2i^+3j^+ #955;2i^ 3j^+6k^ has direction ratios (2, minus;3, ...

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Standard XII

Mathematics

Equation of a Plane Parallel to a Given Plane

Find the vect...

Question

Find the vector equation of a line passing through the point (2, 3, 2) and parallel to the line $\vec{r} = (- 2 \hat{i} + 3 \hat{j}) + λ (2 \hat{i} - 3 \hat{j} + 6 \hat{k}) .$ Also find the distance between these lines.

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Solution

Vector equation of a line passing through (2, 3, 2) and parallel to the line $\vec{r} = (- 2 \hat{i} + 3 \hat{j}) + λ (2 \hat{i} - 3 \hat{j} + 6 \hat{k})$ has direction ratios (2, −3, 6)
Hence, its equation is

$\vec{r} = (2 \hat{i} + 3 \hat{j} + 2 \hat{k}) + λ (2 \hat{i} - 3 \hat{j} + 6 \hat{k})$
Hence, distance between $\vec{r} = {\vec{a}}_{1} + λ {\vec{b}}_{1} and \vec{r} = {\vec{a}}_{2} + μ {\vec{b}}_{1} = {\vec{a}}_{2} + μ {\vec{b}}_{1} is d = |\frac{({\vec{a}}_{1} - {\vec{a}}_{2}) \times \vec{b}}{|\vec{b}|}|$
$Here, {\vec{a}}_{1} = - 2 \hat{i} + 3 \hat{j}, {\vec{a}}_{2} = (2 \hat{i} + 3 \hat{j} + 2 \hat{k}) and \vec{b} = (2 \hat{i} - 3 \hat{j} + 6 \hat{k}) i . e . d = \frac{|(- 4 \hat{i} - 2 \hat{k}) \times (2 \hat{i} - 3 \hat{j} + 6 \hat{k})|}{\sqrt{{(2)}^{2} + {(- 3)}^{2} + {(6)}^{2}}} |({\vec{a}}_{1} - {\vec{a}}_{2}) \times \vec{b}| = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - 4 & 0 & - 2 \\ 2 & - 3 & 6 \end{matrix}| = \hat{i} (- 6) - \hat{j} (- 24 + 4) + \hat{k} (12) i . e |({\vec{a}}_{1} - {\vec{a}}_{2}) \times \vec{b}| = - 6 \hat{i} + 20 \hat{j} + 12 \hat{k} ∴ d = \frac{|- 6 \hat{i} + 20 \hat{j} + 12 \hat{k}|}{\sqrt{4 + 9 + 36}} = \frac{|- 6 \hat{i} + 20 \hat{j} + 12 \hat{k}|}{7} = \frac{2 |- 3 \hat{i} + 10 \hat{j} + 6 \hat{k}|}{7} = \frac{2}{7} \sqrt{9 + 100 + 36} = \frac{2}{7} \times \sqrt{145}$

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Similar questions

Q. Find the distance between the lines l₁ and l₂ given by

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

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from the line passing through the point

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Find the vector equation of a line passing through the point (2, 3, 2) and parallel to the line r→=-2i^+3j^ +λ2i^-3j^+6k^ . Also find the distance between these lines.

Find the vector equation of a line passing through the point (2, 3, 2) and parallel to the line $\vec{r} = (- 2 \hat{i} + 3 \hat{j}) + λ (2 \hat{i} - 3 \hat{j} + 6 \hat{k}) .$ Also find the distance between these lines.