Question 1

Given $|\vec{a} + \vec{b}| = \sqrt{21} \implies |\vec{a} + \vec{b}|^2 = 21 \implies |\vec{a}|^2 + |\vec{b}|^2 + 2\vec{a} \cdot \vec{b} = 21$ ---(i) Given $|\vec{a} - \vec{b}| = 3 \implies |\vec{a} - \vec{b}|^2 = 9 \implies |\vec{a}|^2 + |\vec{b}|^2 - 2\vec{a} \cdot \vec{b} = 9$ ---(ii) Adding (i) and (ii): $2(|\vec{a}|^2 + |\vec{b}|^2) = 30 \implies |\vec{a}|^2 + |\vec{b}|^2 = 15$. Subtracting (ii) from (i): $4\vec{a} \cdot \vec{b} = 12 \implies \vec{a} \cdot \vec{b} = 3$. Since $\vec{a}$ is perpendicular to $(\vec{a} - \vec{b})$, we have $\vec{a} \cdot (\vec{a} - \vec{b}) = 0 \implies |\vec{a}|^2 - \vec{a} \cdot \vec{b} = 0 \implies |\vec{a}|^2 = 3$. Then $|\vec{b}|^2 = 15 - 3 = 12$. Area of triangle $OPR = \frac{1}{2} |\vec{OP} \times \vec{OR}| = \frac{1}{2} |\vec{a} \times (\vec{a} + \vec{b})| = \frac{1}{2} |\vec{a} \times \vec{a} + \vec{a} \times \vec{b}| = \frac{1}{2} |\vec{a} \times \vec{b}|$. Now, $|\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2 = (3)(12) - (3)^2 = 36 - 9 = 27$. So, $|\vec{a} \times \vec{b}| = \sqrt{27} = 3\sqrt{3}$. Area of $\triangle OPR = \frac{3\sqrt{3}}{2}$.