Find the angle between the line r = (i + 2j - k) + λ(i - j + k) and the plane r. (2i - j + k) = 4. ​

Question

Find the angle between the line \(\bar{r}\) = (\(\hat{i}\) + 2\(\hat{j}\) - \(\hat{k}\)) + λ(\(\hat{i}\) - \(\hat{j}\) + \(\hat{k}\)) and the plane \(\bar{r}\). (2\(\hat{i}\) - \(\hat{j}\) + \(\hat{k}\)) = 4.

Answer 1

Given \(\bar{r}\) = (\(\hat{i}\) + 2\(\hat{j}\) - \(\hat{k}\)) + λ(\(\hat{i}\) - \(\hat{j}\) + \(\hat{k}\)) and \(\bar{r}\). (2\(\hat{i}\) - \(\hat{j}\) + \(\hat{k}\)) = 4

To find – The angle between the line and the plane 

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

Direction ratios of the normal of the plane = (2, - 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\((\frac{axa'+bxb'+cxc'}{\sqrt{a^2+b^2+c^2}\sqrt{a^{'^{2}}+b^{'^2}+c^{'^2}}})\)

The angle between the line and the plane