To solve this problem, let's go step-by-step.

1. Given Force Vector $\vec{F}$: $\vec{F} = 3\hat{i} + 4\hat{j}$ This force vector acts at point $A(2, 9)$ with components $3$ in the $i$-direction and $4$ in the $j$-direction.

2. Direction Vector $\vec{BA}$: We need to find the vector $\vec{BA}$ from point $B(3, 7)$ to point $A(2, 9)$. This is calculated as: $\vec{BA} = (2 - 3)\hat{i} + (9 - 7)\hat{j} = -1\hat{i} + 2\hat{j}$

3. Finding the Tangent of the Angle between $\vec{BA}$ and $\vec{F}$: The tangent of the angle $\theta$ between two vectors $\vec{A}$ and $\vec{B}$ can be given by: $\tan \theta = \frac{|\vec{A} \times \vec{B}|}{\vec{A} \cdot \vec{B}}$

   • Dot Product $\vec{BA} \cdot \vec{F}$: $\vec{BA} \cdot \vec{F} = (-1)(3) + (2)(4) = -3 + 8 = 5$
   • Cross Product Magnitude $|\vec{BA} \times \vec{F}|$: $|\vec{BA} \times \vec{F}| = |(-1)(4) - (2)(3)| = |-4 - 6| = |-10| = 10$

   Now, we substitute these values to find $\tan \theta$: $\tan \theta = \frac{10}{5} = 2$

Answer:

The correct answer is (a) $2$.

Would you like further explanation on any step?

Here are five questions for practice:

1. What would the tangent of the angle be if the points $A$ and $B$ were swapped?
2. How would the solution change if $\vec{F}$ was given as $5\hat{i} + 12\hat{j}$?
3. What is the physical significance of the angle between $\vec{BA}$ and $\vec{F}$?
4. How do we find the magnitude of a cross product for two 2D vectors?
5. How would you calculate the angle itself instead of just its tangent?

Tip: To find the angle between two vectors, the dot product and cross product are very useful tools in both 2D and 3D vector problems.