‘Shanta Enterprises’ was in the business of manufacturing solar panels for the last fifteen years. With their expertise now they were able to manufacture good quality solar panels at lower cost. As a result ‘Shanta Enterprises’ was flooded with orders and was able to generate a good profit margin.
To motivate the employees the company decided to compensate some of them for their hard work. Arvind and Yogesh, who were working on contract basis, were absorbed permanently in the company. Both were happy as now there was stability about their future income. Ekta, a regular employee of the company, working as Research and Development Head, was awarded the ‘Best Employee of the year Award’.
Three friends A, B, and C move out from the same location O at the same time in three different directions to reach their destinations. They move out on straight paths and decide that A and B after reaching their destinations will meet up with C at his pre-decided destination, following straight paths from A to C and B to C in such a way that \( \overrightarrow{OA} = \hat{i}, \overrightarrow{OB} = \hat{j} \), and \( \overrightarrow{OC} = 5 \hat{i} - 2 \hat{j} \), respectively.
Based upon the above information, answer the following questions:
(i) Complete the given figure to explain their entire movement plan along the respective vectors.}
(ii) Find vectors \( \overrightarrow{AC} \) and \( \overrightarrow{BC} \).}
(iii) (a) If \( \overrightarrow{a} = 2 \hat{i} - \hat{j} + 4 \hat{k} \), distance of O to A is 1 km, and from O to B is 2 km, then find the angle between \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \). Also, find \( | \overrightarrow{a} \times \overrightarrow{b} | \).}
(iii) (b) If \( \overrightarrow{a} = 2 \hat{i} - \hat{j} + 4 \hat{k} \), find a unit vector perpendicular to \( (\overrightarrow{a} + \overrightarrow{b}) \) and \( (\overrightarrow{a} - \overrightarrow{b}) \).