Find the vector and Cartesian equations of the plane passing through the origin and parallel to the vectors (i + j - k) and (3i - k). ​

Question

Find the vector and Cartesian equations of the plane passing through the origin and parallel to the vectors (\(\hat{i}\) + \(\hat{j}\) - \(\hat{k}\)) and (3\(\hat{i}\) - \(\hat{k}\)).

Answer 1

Given - \(\bar{r}\) = \(\hat{i}\) + \(\hat{j}\) - \(\hat{k}\) and \(\bar{r}\) = 3\(\hat{i}\) - \(\hat{k}\) are two lines to which a plane is parallel and it passes through the origin. 

To find – The equation of the plane 

Tip – A plane parallel to two vectors will have its normal in a direction perpendicular to both the vectors, which can be evaluated by taking their cross product

The plane passes through origin (0, 0, 0). Formula to be used – If a line passes through the point (a, b, c) and has the direction ratios as (a’, b’, c’), then its vector equation is given by  \(\bar{r}\) = (a\(\hat{i}\) + b\(\hat{j}\) + c\(\hat{k}\)) + λ (a'\(\hat{i}\) + b'\(\hat{j}\) + c''\(\hat{k}\)) where λ is any scalar constant 

The required plane will be

The Cartesian equation : x + 2y + 3z = 0