Nishi had gone to a grocery store to make routine purchases. On reaching home, as she took out Binx tomato chips packet from the bag to give it to her son, she felt that it was underweight. She checked its weight on the kitchen weighing scale and found that it weighed 60 grams whereas the label on the chips packet mentioned the weight of the packet as 100 grams. She approached the manufacturer and complained about it. The manufacturer offered her a gift hamper and requested her not to disclose this to anyone. Nishi refused to accept the gift hamper and took the issue to a redressal agency.
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 \( (\overrightarrow{a} - \overrightarrow{b}) \).