Find the angle between the line r = (2i - j + 3k) + λ(3i - j + 2k) and the plane r. (i + j + k) = 3. ​

Question

Find the angle between the line \(\bar{r}\) = (2\(\hat{i}\) - \(\hat{j}\) + 3\(\hat{k}\)) + λ(3\(\hat{i}\) - \(\hat{j}\) + 2\(\hat{k}\)) and the plane \(\bar{r}\). (\(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\)) = 3.

Answer 1

Given  \(\bar{r}\) = (2\(\hat{i}\) - \(\hat{j}\) + 3\(\hat{k}\)) + λ(3\(\hat{i}\) - \(\hat{j}\) + 2\(\hat{k}\)) and \(\bar{r}\). (\(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\)) = 3

To find – The angle between the line and the plane 

Direction ratios of the line = (3, - 1, 2) 

Direction ratios of the normal of the plane = (1, 1, 1) 

Formula to be used – If (a, b, c) be the direction ratios of a line and (a’, b’, c’) be the direction ratios of the normal to the plane, then, the angle between the two is given by sin^-1\((\cfrac{axa'+bxb'+cxc'}{\sqrt{a^2+b^2+c^2}\sqrt{a^{'^{2}}+b^{'^2}+c^{'^2}}})\)

The angle between the line and the plane