1. **State the problem:** We have two velocity vectors for projectiles fired from cannons: $$\vec{v} = 200\mathbf{i} + 150\mathbf{j}$$ and $$\vec{u} = 250\mathbf{i} + 100\mathbf{j}$$ We need to find: (a) The magnitude of $$\vec{v}$$. (b) The angle $$\vec{v}$$ makes with the horizontal axis. (i) The angle between $$\vec{v}$$ and $$\vec{u}$$. (ii) Whether the two projectiles are fired in the same direction. 2. **Find the magnitude of $$\vec{v}$$:** The magnitude of a vector $$\vec{v} = v_x\mathbf{i} + v_y\mathbf{j}$$ is given by: $$ |\vec{v}| = \sqrt{v_x^2 + v_y^2} $$ For $$\vec{v} = 200\mathbf{i} + 150\mathbf{j}$$: $$ |\vec{v}| = \sqrt{200^2 + 150^2} = \sqrt{40000 + 22500} = \sqrt{62500} = 250 $$ 3. **Calculate the angle $$\theta$$ made by $$\vec{v}$$ with the horizontal axis:** The angle $$\theta$$ is given by: $$ \theta = \tan^{-1}\left(\frac{v_y}{v_x}\right) $$ Substitute values: $$ \theta = \tan^{-1}\left(\frac{150}{200}\right) = \tan^{-1}(0.75) \approx 36.87^\circ $$ 4. **Find the angle between $$\vec{v}$$ and $$\vec{u}$$:** The angle $$\phi$$ between two vectors $$\vec{v}$$ and $$\vec{u}$$ is given by: $$ \cos \phi = \frac{\vec{v} \cdot \vec{u}}{|\vec{v}||\vec{u}|} $$ Calculate the dot product: $$ \vec{v} \cdot \vec{u} = (200)(250) + (150)(100) = 50000 + 15000 = 65000 $$ Calculate magnitudes: $$ |\vec{v}| = 250 \quad (from\ step\ 2) $$ $$ |\vec{u}| = \sqrt{250^2 + 100^2} = \sqrt{62500 + 10000} = \sqrt{72500} \approx 269.26 $$ Calculate $$\cos \phi$$: $$ \cos \phi = \frac{65000}{250 \times 269.26} = \frac{65000}{67315} \approx 0.9655 $$ Calculate $$\phi$$: $$ \phi = \cos^{-1}(0.9655) \approx 15.28^\circ $$ 5. **Determine if the projectiles are fired in the same direction:** Since the angle between the two velocity vectors is $$15.28^\circ$$, which is not zero, the projectiles are not fired in exactly the same direction. **Final answers:** (a) Magnitude of $$\vec{v}$$ is $$250$$ m/s. (b) Angle of $$\vec{v}$$ with horizontal is approximately $$36.87^\circ$$. (i) Angle between $$\vec{v}$$ and $$\vec{u}$$ is approximately $$15.28^\circ$$. (ii) The projectiles are not fired in the same direction.