Random Mathematics - Vectors And Linear Programming Practice

Case Study 1

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 on straight paths and decide that A and B will meet C at his destination following straight paths from A to C and B to C such that $\overrightarrow{OA} = \vec{a}$, $\overrightarrow{OB} = \vec{b}$ and $\overrightarrow{OC} = 5\vec{a} - 2\vec{b}$.

Based upon the above information, answer the following questions:

(i) Complete the given figure to explain their entire movement plan along the respective vectors. [1 mark]

(ii) Find vectors $\overrightarrow{AC}$ and $\overrightarrow{BC}$. [1 mark]

(iii)(a) If $\vec{a} \cdot \vec{b} = 1$, distance OA = 1 km and distance OB = 2 km, then find the angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$. Also find $|\vec{a} \times \vec{b}|$. [2 marks]

OR

(iii)(b) If $\vec{a} = 2\hat{i} - \hat{j} + 4\hat{k}$ and $\vec{b} = \hat{j} - \hat{k}$, then find a unit vector perpendicular to both $(\vec{a} + \vec{b})$ and $(\vec{a} - \vec{b})$. [2 marks]