Vectors: EJC 2024 Promo Q5

(a)

Required length of projection is $$\begin{aligned} \left|\,|\mathbf{a}|\cos\frac{5\pi}{6}\,\right| &= \left|\sqrt{2}\times\left(-\frac{\sqrt{3}}{2}\right)\right| = \frac{\sqrt{6}}{2}. \end{aligned}$$

(b)

$$\begin{aligned} |3\mathbf{a} + 2\mathbf{b}|^2 &= (3\mathbf{a} + 2\mathbf{b})\cdot(3\mathbf{a} + 2\mathbf{b})\\ &= (3\mathbf{a})\cdot(3\mathbf{a})+(2\mathbf{b})\cdot(3\mathbf{a}) +(3\mathbf{a})\cdot(2\mathbf{b})+(2\mathbf{b})\cdot(2\mathbf{b})\\ &= 9|\mathbf{a}|^2 + 12\mathbf{a}\cdot\mathbf{b} + 4|\mathbf{b}|^2\\ &= 9(2) + 12(\sqrt{2})(\sqrt{6})\left(-\frac{\sqrt{3}}{2}\right) + 4(6)\\ &= 6. \end{aligned}$$ So $|3\mathbf{a} + 2\mathbf{b}| = \sqrt{6}$.