Name: The Shortest Distance Between Two Skew Lines
Uploaded: 2019-07-19
Duration: 9 min 21 s
Description: Skew lines are the lines that do not intersect and not parallel. The shortest distance between two skew lines lies along the line, which is perpendicular to...

Question

Find the equations of the two lines through the origin which intersect the line

\frac{x - 3}{2} = \frac{y - 3}{1} = \frac{z}{1}

at angle of

\frac{π}{3}

each

Accepted Answer

The coordinates of any point on the line $\frac{x - 3}{2} = \frac{y - 3}{1} = \frac{z}{1}$ are given by $\frac{x - 3}{2} = \frac{y - 3}{1} = \frac{z}{1} = λ$

So, let the coordinates of $A$ be $(2 λ + 3, λ + 3, λ)$ .

Let, the line through $O (0, 0, 0)$ and making an angle $\frac{π}{3}$ with the given line be along $O A .$

The direction ratios of a line joining the points $(x_{1}, y_{1}, z_{1})$ and $(x_{2}, y_{2}, z_{2})$ are proportional to $x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1}$ .

Then, its direction ratios are proportional to

$2 λ + 3 - 0, λ + 3 - 0, λ - 0$ or $2 λ + 3, λ + 3, λ$

The direction ratio of the given line are proportional to $2, 1, 1 .$

It is given that the angle between the given line and the line along $O A$ is $\frac{π}{3}$ and we know that the angle between two lines whose direction ratios are $a_{1}, b_{1}, c_{1}$ and $a_{2}, b_{2}, c_{2}$ is $\cos θ = \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}} \sqrt{a_{2}^{2} + b_{2}^{2} + c_{2}^{2}}}$

$\Rightarrow \cos \frac{π}{3} = \frac{(2 λ + 3) \times 2 + (λ + 3) \times 1 + λ \times 1}{\sqrt{{(2 λ + 3)}^{2} + {(λ + 3)}^{2} + λ^{2}} \sqrt{2^{2} + 1^{2} + 1^{2}}}$

$\Rightarrow \frac{1}{2} = \frac{6 λ + 9}{\sqrt{6 λ^{2} + 18 λ + 18} \sqrt{6}}$

$\Rightarrow (6 λ^{2} + 18 λ + 18) = 6 {(2 λ + 3)}^{2}$

$\Rightarrow 6 λ^{2} + 18 λ + 18 = 24 λ^{2} + 72 λ + 54$

$\Rightarrow 18 λ^{2} + 54 λ + 36 = 0$

$\Rightarrow λ^{2} + 3 λ + 2 = 0$

$∴ λ = - 1, - 2$

$\frac{x - 3}{2} = \frac{y - 3}{1} = \frac{z}{1}$

Putting these values of $λ$ in the coordinates of $A$ , i.e., $2 λ + 3, λ + 3, λ,$ we find the coordinates of $A$ and $B$

i.e., $A (1, 2, - 1)$ and $B (- 1, 1, - 2) .$

Also, we know that the equation of a line passing through the points $(x_{1}, y_{1}, z_{1})$ and $(x_{2}, y_{2}, z_{2})$ is $\frac{x - x_{1}}{x_{2} - x_{1}} = \frac{y - y_{1}}{y_{2} - y_{1}} = \frac{z - z_{1}}{z_{2} - z_{1}}$

Hence, the equation of the lines $O A$ and $O B$ are

$\Rightarrow \frac{x - 0}{1 - 0} = \frac{y - 0}{2 - 0} = \frac{z - 0}{- 1 - 0}$ and $\frac{x - 0}{- 1 - 0} = \frac{y - 0}{1 - 0} = \frac{z - 0}{- 2 - 0}$

$\Rightarrow \frac{x}{1} = \frac{y}{2} = \frac{z}{- 1}$ and $\frac{x}{- 1} = \frac{y}{1} = \frac{z}{- 2}$ .

Find the equations of the two lines through the origin which intersect the line $\frac{x - 3}{2} = \frac{y - 3}{1} = \frac{z}{1}$ at angle of $\frac{π}{3}$ each

Important Questions on Three Dimensional Geometry

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Find the equations of the two lines through the origin which intersect the line x-32=y-31=z1 at angle of π3 each

Important Questions on Three Dimensional Geometry

Learn and Prepare for any exam you want !

Find the equations of the two lines through the origin which intersect the line $\frac{x - 3}{2} = \frac{y - 3}{1} = \frac{z}{1}$ at angle of $\frac{π}{3}$ each