The positions of a moving particle at two instants \( 3^{\text {rd }} \) second and \( 8^{\text {th }} \) seconds was at \( A(7,2) \) and \( B(4,6) \) respectively, then the average velocity of the particle \( \vec{v}_{A}=\cdots \) \( \begin{array}{llll}\text { (a) } 3 \vec{i}-4 \vec{j} & \text { (b) }-3 \vec{i}+4 \vec{j} & \text { (c) } \frac{3}{5} \vec{i}-\frac{4}{5} \vec{j} & \text { (d) } \frac{-3}{5} \vec{i}+\frac{4}{5} \vec{j}\end{array} \)

To find the average velocity of the particle between the two points \( A(7,2) \) and \( B(4,6) \), we can use the formula for average velocity: \[ \vec{v}_{avg} = \frac{\Delta \vec{x}}{\Delta t} = \frac{(x_B - x_A) \vec{i} + (y_B - y_A) \vec{j}}{t_B - t_A} \] Here, \( \Delta x = 4 - 7 = -3 \), \( \Delta y = 6 - 2 = 4 \), and \( \Delta t = 8 - 3 = 5 \). Plugging these into the formula gives: \[ \vec{v}_{avg} = \frac{(-3) \vec{i} + 4 \vec{j}}{5} = -\frac{3}{5} \vec{i} + \frac{4}{5} \vec{j} \] Therefore, the average velocity of the particle is: \(-\frac{3}{5} \vec{i} + \frac{4}{5} \vec{j}\) (Option d). The correct answer is \( \text{(d) } \frac{-3}{5} \vec{i}+\frac{4}{5} \vec{j} \). Now, let’s spice things up with some fun facts! Did you know that understanding average velocity is essential not just in physics but also in fields like sports and animation? Coaches analyze average speeds of players so they can fine-tune strategies, while animators use it to create fluid movements that appear realistic! Also, if you're fascinated by graphs, try plotting the points \( A \) and \( B \) on a Cartesian plane. You'll see a straight line connecting them—this visual representation makes understanding motion much easier! Plus, it’s a fun activity that solidifies your grasp on the concept.