Show that the line r = (2i - 2j + 3k) + λ(i - j + 4k) is parallel to the plane r.( i+ j5 + k) = 7 ​

Question

Show that the line \(\bar{r}\) = (2\(\hat{i}\) - 2\(\hat{j}\) + 3\(\hat{k}\)) + λ(\(\hat{i}\) - \(\hat{j}\) + 4\(\hat{k}\)) is parallel to the plane \(\bar{r}\).(\(\hat{i}\) + 5\(\hat{j}\) + \(\hat{k}\)) = 7

Answer 1

Given : 

Equation of plane: \(\bar{r}\).(\(\hat{i}\) + 5\(\hat{j}\) + \(\hat{k}\)) = 7

Equation of a line :

Therefore, a vector normal to the plane is perpendicular to the vector parallel to the line. 

Hence, the given line is parallel to the given plane.