1. **State the problem:** We are given the acceleration vector formula $$\vec{a} = \frac{\vec{v}_f - \vec{v}_i}{t}$$ and values $$\vec{v}_f = 5.2\ \text{m/s},\ t = 1.50\ \text{s},\ \vec{a} = 23.00\ \text{m/s}^2$$. We need to solve for the initial velocity vector $$\vec{v}_i$$. 2. **Formula and explanation:** The formula relates acceleration, initial velocity, final velocity, and time. Rearranging to solve for $$\vec{v}_i$$: $$ \vec{v}_i = \vec{v}_f - \vec{a} \times t $$ This means the initial velocity equals the final velocity minus the product of acceleration and time. 3. **Substitute the known values:** $$ \vec{v}_i = 5.2 - 23.00 \times 1.50 $$ 4. **Calculate the product:** $$ 23.00 \times 1.50 = 34.5 $$ 5. **Calculate initial velocity:** $$ \vec{v}_i = 5.2 - 34.5 = -29.3\ \text{m/s} $$ 6. **Interpretation:** The negative sign indicates the initial velocity vector is in the opposite direction to the final velocity vector. **Final answer:** $$\vec{v}_i = -29.3\ \text{m/s}$$