$3$Three friends $\displaystyle \mathrm{A}, \mathrm{B}$ 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 predecided destination, following straight paths from A to C and B to C in such a way that $\displaystyle \overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{a}}$, $\displaystyle \overrightarrow{\mathrm{OB}}=\overrightarrow{\mathrm{b}}$ and $\displaystyle \overrightarrow{\mathrm{OC}}=5 \overrightarrow{\mathrm{a}}-2 \overrightarrow{\mathrm{~b}}$ respectively. 2025_CBSE_Mathematics_Set_Region7_Subset_Set2_Q37___p11_8

2025_CBSE_Mathematics_Set_Region7_Subset_Set2_Q37___p11_8

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 $\displaystyle \overrightarrow{\mathrm{AC}}$ and $\displaystyle \overrightarrow{\mathrm{BC}}$.(iii)(a) If $\displaystyle \vec{\mathrm{a}} \cdot \vec{\mathrm{b}}=1$, distance of O to A is 1 km and that from O to B is 2 km , then find the angle between $\displaystyle \overrightarrow{\mathrm{OA}}$ and $\displaystyle \overrightarrow{\mathrm{OB}}$. Also, find $\displaystyle |\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|$.

OR

(iii) (b) If $\displaystyle \vec{\mathrm{a}}=2 \hat{i}-\hat{j}+4 \hat{\mathrm{k}}$ and $\displaystyle \vec{\mathrm{b}}=\hat{j}-\hat{\mathrm{k}}$, then find a unit vector perpendicular to $\displaystyle (\vec{\mathrm{a}}+\vec{\mathrm{b}})$ and $\displaystyle (\vec{\mathrm{a}}-\vec{\mathrm{b}})$. Case Study - 3

Mathematics · Vector Algebra