Let \(\left|\vec{A_1}\right|=3,\left|\vec{A_2}\right|=5\) and \(\left|\vec{A_1}+\vec{A_2}\right|=5.\) The value of \(\left(\vec{2 A_1}+3 \vec{A_2}\right) \cdot\left(3 \vec{A_1}-2 \vec{A_2}\right) \) is:

A balloon is rising vertically above a fixed point \(A\) on the ground. A girl at point \(B,\) located \(d\) metres from \(A,\) observes the balloon at a height \(h_1\)​ and sees it at an angle of \(45^\circ\) with respect to the vertical. After the balloon ascends an additional height \(h_2,\) the girl walks \(2.464d\) metres further away from point \(A\) to reach point \(C.\) From this new position, she observes the balloon at an angle of \(60^\circ\) with the vertical.  If \(\tan 30^{\circ}=0.5774,\) then what is the value of \(h_2\)​ in terms of \(d \text{?}\)

A particle moving in the xy plane experiences a velocity-dependent force \(\vec F= k\left(v_y\hat i +v_x \hat j\right)\) where \(v_x\) and \(v_y\) are the \(x\) and \(y\) components of its velocity \(\vec{v}\). If \(\vec{a}\) is the acceleration of the particle, then which of the following statements is true for the particle? 

In a regular octagon \({ABCDEFGH},\) all sides are equal in length. The position vector of point \(A\) with respect to the center \(O\) of the octagon is given by: \(\overrightarrow{{AO}}=2 \hat{{i}}+3 \hat{{j}}-4 \hat{{k}}.\)
What is the value of the vector sum: \(\overrightarrow{{AB}}+\overrightarrow{{AC}}+\overrightarrow{{AD}}+\overrightarrow{{AE}}+\overrightarrow{{AF}}+\overrightarrow{{AG}}+\overrightarrow{{AH}} ~\text{?}\)

If \(\vec{P} \times \vec{Q}=\vec{Q} \times \vec{P},\) the angle between \(\vec{P}\) and \(\vec{Q}\) is \(\theta\) \((0^\circ<\theta<360^\circ),\) then the value of \(\theta\) will be:
1. \(30^\circ\)
2. \(60^\circ\)
3. \(90^\circ\)
4. \(180^\circ\)