​ Find the vector and Cartesian forms of the equations of the plane containing the two lines r = (i + 2j - 4k) + λ(2i + 3j + 6k)

Question

Find the vector and Cartesian forms of the equations of the plane containing the two lines \(\bar{r}\) = (\(\hat{i}\) + 2\(\hat{j}\) - 4\(\hat{k}\)) + λ(2\(\hat{i}\) + 3\(\hat{j}\)  + 6\(\hat{k}\)) and \(\bar{r}\) = (9\(\hat{i}\) + 5\(\hat{j}\) + \(\hat{k}\)) + \(\mu\) (-2\(\hat{i}\) + 3\(\hat{j}\) + 8\(\hat{k}\)).

Answer 1

Given : Equations of lines 

To Find : Equation of plane. 

Formulae : 

1) Cross Product :

If two lines \(\bar{r_1}\) = \(\bar{a_1}\) + A\(\bar{b_1}\) & \(\bar{r_2}\) + \(\bar{a_2}\) + A\(\bar{b_2}\) are coplanar then equation of the plane containing them is

Equation of plane containing lines \(\bar{r_1}\) & \(\bar{r_2}\) is

= 6x – 28y + 12z 

Therefore, equation of plane is 

6x – 28y + 12z = -98 

6x – 28y + 12z + 98 = 0 

This Cartesian equation of plane.