Discuss the properties of scalar and vector products.

Derive the equations of motion for a particle
(a) falling vertically
(b) projected vertically.

Derive the expression for centripetal acceleration.

Two vectors \(\vec A\) and \(\vec B\) of magnitude 5 units and 7 units respectively make an angle 60° with each other as shown below. Find the magnitude of the resultant vector and its direction with respect to 7 unit the vector \(\vec A\).

Suppose an object is thrown with initial speed 10 ms^-1 at an angle \(\frac{\pi}{4}\) with the horizontal, what is the range covered? Suppose the same object is thrown similarly in the Moon, will there be any change in the range? If yes, what is the change? (The acceleration due to gravity in the Moon \(g_{moon}=\frac{1}{6}g\)).