If equations of two non intersecting lines are -x-x1-l1 - y-y1-m1 - z-z1-n1 x-x2-l2 - y-y2-m2 - z-z2-n2 then the shortest distance between them will be -

Let’s rewrite the first line’s equation x x_1/l_1 = y y_1/m_1 = z z_1/n_1 = r_1 (say) Any point on first line( let’s call it “P” ) can be written as (x_1+l_1. ...

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Byju's Answer

Standard XIII

Mathematics

Shortest Distance between Two Skew Lines

If equations ...

Question

If equations of two non intersecting lines are -
$\frac{x - x_{1}}{l_{1}} = \frac{y - y_{1}}{m_{1}} = \frac{z - z_{1}}{n_{1}}$ & $\frac{x - x_{2}}{l_{2}} = \frac{y - y_{2}}{m_{2}} = \frac{z - z_{2}}{n_{2}}$ then the shortest distance between them will be -

\frac{1}{\sqrt{Σ (l_{1} m_{2} - l_{2} m_{1})^{2}}} ∣ ∣ ∣ ∣ \begin{matrix} x 1 & y 1 & z 1 l 1 & m 1 & n 1 l 2 & m 2 & n 2 \end{matrix} ∣ ∣ ∣ ∣

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\frac{1}{\sqrt{Σ (l_{1} m_{2} - l_{2} m_{1})^{2}}} ∣ ∣ ∣ ∣ \begin{matrix} x 2 - x 1 & y 2 - y 1 & z 2 - z 1 l 1 & m 1 & n 1 l 2 & m 2 & n 2 \end{matrix} ∣ ∣ ∣ ∣

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\frac{1}{\sqrt{Σ (l_{1} m_{2} - l_{2} m_{1})^{2}}} ∣ ∣ ∣ ∣ \begin{matrix} x 2 & y 2 & z 2 l 1 & m 1 & n 1 l 2 & m 2 & n 2 \end{matrix} ∣ ∣ ∣ ∣

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Solution

The correct option is B $\frac{1}{\sqrt{Σ (l_{1} m_{2} - l_{2} m_{1})^{2}}} ∣ ∣ ∣ ∣ \begin{matrix} x 2 - x 1 & y 2 - y 1 & z 2 - z 1 l 1 & m 1 & n 1 l 2 & m 2 & n 2 \end{matrix} ∣ ∣ ∣ ∣$
Let’s rewrite the first line’s equation -
$\frac{x - x_{1}}{l_{1}} = \frac{y - y_{1}}{m_{1}} = \frac{z - z_{1}}{n_{1}} = r_{1}$ (say)
Any point on first line( let’s call it “P” ) can be written as $(x_{1} + l_{1} . r_{1}, y_{1} + m_{1} . r_{1}, z_{1} + n_{1} . r_{1})$
Similarly equation of second line can be rewritten as -
$\frac{x - x_{2}}{l_{2}} = \frac{y - y_{2}}{m_{2}} = \frac{z - z_{2}}{n_{2}} = r_{2}$ (say)
And any point on second line( let’s call it “Q” )can be written as
$(x_{2} + l_{2} . r_{2}, y_{2} + m_{2} . r_{2}, z_{2} + n_{2} . r_{2})$
Direction ratios of PQ will be -
$(x_{1} + l_{1} . r_{1} - (x_{2} + l_{2} . r_{2}), y_{1} + m_{1} . r_{1} - (y_{2} + m_{2} . r_{2}), z_{1} + n_{1} . r_{1} - (z_{2} + n_{2} . r_{2}))$
This line will be the shortest line only if it’s perpendicular to both the given lines.
PQ is perpendicular to the first line then -
$(x_{1} + l_{1} . r_{1} - (x_{2} + l_{2} . r_{2})) . (l_{1}) + (y_{1} + m_{1} . r_{1} - (y_{2} + m_{2} . r_{2})) (m_{1}) (z_{1} + n_{1} . r_{1} - (z_{2} + n_{2} . r_{2})) . (n_{1}) = 0$
Similarly, PQ will be perpendicular to the second line -
$(x_{1} + l_{1} . r_{1} - (x_{2} + l_{2} . r_{2})) . (l_{2}) + (y_{1} + m_{1} . r_{1} - (y_{2} + m_{2} . r_{2})) (m_{2}) (z_{1} + n_{1} . r_{1} - (z_{2} + n_{2} . r_{2})) . (n_{2}) = 0$
On solving these equations we get -
$\frac{1}{\sqrt{Σ (l_{1} m_{2} - l_{2} m_{1})^{2}}} ∣ ∣ ∣ ∣ \begin{matrix} x 2 - x 1 & y 2 - y 1 & z 2 - z 1 l 1 & m 1 & n 1 l 2 & m 2 & n 2 \end{matrix} ∣ ∣ ∣ ∣$

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Similar questions

Q. If equations of two non intersecting lines are -

\frac{x - x_{1}}{l_{1}} = \frac{y - y_{1}}{m_{1}} = \frac{z - z_{1}}{n_{1}}

\frac{x - x_{2}}{l_{2}} = \frac{y - y_{2}}{m_{2}} = \frac{z - z_{2}}{n_{2}}

then the shortest distance between them will be -

Q. Assertion :Three concurrent lines with direction cosines

l_{1}, m_{1}, n_{1}; l_{2}, m_{2}, n_{2}; l_{3}, m_{3}, n_{3}

are coplanar if

∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0

Reason: Lines

\frac{x - x_{1}}{l_{1}} = \frac{y - y_{1}}{m_{1}} = \frac{z - z_{1}}{n_{1}}, \frac{x - x_{2}}{l_{2}} = \frac{y - y_{2}}{m_{2}} = \frac{z - z_{2}}{n_{2}}

are coplanar if

∣ ∣ ∣ ∣ \begin{matrix} x - x_{1} & y - y_{1} & z - z_{1} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} \end{matrix} ∣ ∣ ∣ ∣ = 0

Q. Assertion :If the lines

\frac{x - x_{1}}{a_{1}} = \frac{y - y_{1}}{b_{1}} = \frac{z - z_{1}}{c_{1}}

and

\frac{x - x_{2}}{a_{2}} = \frac{y - y_{2}}{b_{2}} = \frac{z - z_{2}}{c_{2}}

are coplanar then

∣ ∣ ∣ ∣ \begin{matrix} x_{1} & y_{1} & z_{1} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} x_{2} & y_{2} & z_{2} a_{1} & b_{1} & c_{1} a_{2} & b_{1} & c_{1} \end{matrix} ∣ ∣ ∣ ∣

Reason: If two lines are coplanar then shortest distance between them is zero

Q. The shortest distance between lines

\frac{x - x_{1}}{a_{1}} = \frac{y - y_{1}}{b_{1}} = \frac{z - z_{1}}{c_{1}}

and

\frac{x - x_{2}}{a_{2}} = \frac{y - y_{2}}{b_{2}} = \frac{z - z_{2}}{c_{2}}

is given by

Q. If two lines with direction cosines

l_{1}, m_{1}, n_{1}

and

l_{2}, m_{2}, n_{2}

intersect, then the angle between them would be -

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If equations of two non intersecting lines are - x−x1l1=y−y1m1=z−z1n1 & x−x2l2=y−y2m2=z−z2n2 then the shortest distance between them will be -

If equations of two non intersecting lines are -
$\frac{x - x_{1}}{l_{1}} = \frac{y - y_{1}}{m_{1}} = \frac{z - z_{1}}{n_{1}}$ & $\frac{x - x_{2}}{l_{2}} = \frac{y - y_{2}}{m_{2}} = \frac{z - z_{2}}{n_{2}}$ then the shortest distance between them will be -