question_answer

Two groups of students representing 'SAVE MOTHER EARTH' and 'GO GREEN' are standing on two planes,, represented by the equations. \[\vec{r}\cdot (\hat{i}+\hat{j}+2\hat{k})=5\] and \[\vec{r}\cdot (2\hat{i}-\hat{j}+\hat{k})=8.\] What is the angle between the planes? Name few activities which should be taken up to save mother earth.

Answer:

Given, equation of two planes are                         \[\vec{r}\cdot (\hat{i}+\hat{j}+2\hat{k})=5\]                  ?(i)             and       \[\vec{r}\cdot (2\hat{i}-\hat{j}+\hat{k})=8\]                    ?(ii) By comparing the equation of planes (i) and (ii) with \[\vec{r}\cdot {{\vec{n}}_{1}}={{d}_{1}}\] and \[\vec{r}\cdot {{\vec{n}}_{2}}={{d}_{2}}\] is given by             \[\cos \theta =\frac{{{{\vec{n}}}_{1}}\cdot {{{\vec{n}}}_{2}}}{|{{{\vec{n}}}_{1}}||{{{\vec{n}}}_{2}}|}\] \[\therefore \]      \[\cos \theta =\frac{(\hat{i}+\hat{j}+2\hat{k})\cdot (2\hat{i}-\hat{j}+\hat{k})}{|(\hat{i}+\hat{j}+2\hat{k})||(2\hat{i}-\hat{j}+\hat{k})|}\]             \[=\frac{2-1+2}{\sqrt{{{(1)}^{2}}+{{(1)}^{2}}+{{(2)}^{2}}\sqrt{{{(2)}^{2}}}+{{(-\,1)}^{2}}+{{(1)}^{2}}}}\]             \[=\frac{3}{\sqrt{6}\sqrt{6}}=\frac{3}{6}\] \[\Rightarrow \]   \[cos\theta =\frac{1}{2}\] \[\Rightarrow \] \[\cos \frac{\pi }{3}\] \[\Rightarrow \] \[\theta =\frac{\pi }{3}\] Hence, angle between two planes is \[\frac{\pi }{3}.\] Values We should be concerned about not wasting natural resources, having development in a planned manner,