1.Prove the vector triple product expansion formula $\vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A} \cdot \vec{C})\vec{B} - (\vec{A} \cdot \vec{B})\vec{C}$.
Sol: Expanding x-component: $[\vec{A} \times (\vec{B} \times \vec{C})]_x = A_y(B_x C_y - B_y C_x) - A_z(B_z C_x - B_x C_z) = B_x(A_y C_y + A_z C_z) - C_x(A_y B_y + A_z B_z)$. Adding and subtracting $A_x B_x C_x$ gives $B_x(\vec{A}\cdot\vec{C}) - C_x(\vec{A}\cdot\vec{B})$. Similarly for y and z components.
2.Find the volume of a parallelepiped formed by vectors $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$, $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$, $\vec{c} = 3\hat{i} - \hat{j} + 2\hat{k}$.
Sol: Volume = $|\vec{a} \cdot (\vec{b} \times \vec{c})| = \begin{vmatrix} 2 & -3 & 4 \\ 1 & 2 & -1 \\ 3 & -1 & 2 \end{vmatrix} = 2(4-1) + 3(2+3) + 4(-1-6) = 6 + 15 - 28 = |-7| = 7\text{ cubic units}$.
3.If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ and $r = |\vec{r}|$, calculate $\nabla r^n$.
Sol: $\nabla r^n = \sum \hat{i} \frac{\partial r^n}{\partial x} = \sum \hat{i} n r^{n-1} \frac{x}{r} = n r^{n-2} (x\hat{i} + y\hat{j} + z\hat{k}) = n r^{n-2} \vec{r}$.
4.Find the work done by a force field $\vec{F} = x^2\hat{i} - xy\hat{j}$ along the straight line from $(0,0)$ to $(1,1)$.
Sol: Along $y=x \implies dy=dx$. $W = \int_0^1 (x^2 dx - x(x) dx) = \int_0^1 0 dx = 0\text{ Joules}$.
