Equation of a plane contains a line

Question

How can we write equation of a plane which contains a line l$_1$ and is parrallel to another line l$_2$ .

I am not getting start , how to do it .

Please help me

Answer 1

Denoting the normalized direction vector of line $l_1$ and $l_2$ as $\vec{d_1}$ and $\vec{d_2}$ respectively, the plane equation can be written as

$ (P(u,v) - P_1) \cdot \vec{n} = 0 $

where $P_1$ is a point on line $l_1$ and $\vec{n}=\vec{d_1}\times\vec{d_2}$ is the normal vector of the plane.

In the case where $\vec{d_1}$ is parallel to $\vec{d_2}$, $\vec{n}$ can be found from $\vec{P_1P_2}$ where $P_2$ is the projected point of $P_1$ on line $l_2$.

Answer 2

Let's say $l_1$ is parallel to vector $\vec v_1$ and $l_2$ is parallel to vector $\vec v_2$

Then the plane you want will be parallel to to both $\vec v_1$ and $\vec v_2$

Pick a point on $l_1$ which has coordinate vector say $\vec v$.

Then that point will lie on the plane.

Hence the plane will have parametric representation $\vec x = \vec v + \alpha \vec v_1 + \beta \vec v_2 $ for real constants $\alpha$ and $\beta$

Answer 3

For writing equation of any plane bassically we need 3 known parameters ,Now here we already have have equation of line which is located in unknown plane ,So from this part we can extract one parameter i.e direction cosine of line .For other paramter we can take any points on line as point on plane ( Line is located on plane therefore any point on line is also on plane ) ,Now the third paramter will come from the extra condition given on the question i.e it is plane is parallel to another line say l2 ,from here we can extract another piece of information i.e another direction cosine . Now assume any any arbitrary point say (x,y,z) lies on plane .Look we have already have another known point from line lies on plane ,so we can write direction cosine using that say known point is (p,q,r) Direction cosine is(x-p)i+(y-q)j+(z-r)k in vector notation
Now dot product of this will be equal to zero with the resultant cross product direction cosine extracted from the known line .That's our final answer