1.If $|\vec{A} + \vec{B}| = |\vec{A} - \vec{B}|$, find the angle between $\vec{A}$ and $\vec{B}$.
Sol: Squaring both sides: $A^2 + B^2 + 2AB\cos\theta = A^2 + B^2 - 2AB\cos\theta \implies 4AB\cos\theta = 0 \implies \cos\theta = 0 \implies \theta = 90^\circ$.
2.Find a unit vector perpendicular to both $\vec{A} = 2\hat{i} + \hat{j} - \hat{k}$ and $\vec{B} = \hat{i} - \hat{j} + 2\hat{k}$.
Sol: $\vec{A} \times \vec{B} = \hat{i}(2-1) - \hat{j}(4+1) + \hat{k}(-2-1) = \hat{i} - 5\hat{j} - 3\hat{k}$. Unit vector = $\pm \frac{\hat{i} - 5\hat{j} - 3\hat{k}}{\sqrt{1 + 25 + 9}} = \pm \frac{\hat{i} - 5\hat{j} - 3\hat{k}}{\sqrt{35}}$.
3.Find the area of a parallelogram whose adjacent sides are $\vec{A} = 3\hat{i} + \hat{j} + 4\hat{k}$ and $\vec{B} = \hat{i} - \hat{j} + \hat{k}$.
Sol: $\vec{A} \times \vec{B} = 5\hat{i} + \hat{j} - 4\hat{k}$. Area = $|\vec{A} \times \vec{B}| = \sqrt{25 + 1 + 16} = \sqrt{42}\text{ sq units}$.
4.A vector $\vec{A}$ makes angles $\alpha, \beta, \gamma$ with x, y, z axes. Prove that $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$.
Sol: Direction cosines $l = A_x/A, m = A_y/A, n = A_z/A$. $l^2 + m^2 + n^2 = \frac{A_x^2 + A_y^2 + A_z^2}{A^2} = \frac{A^2}{A^2} = 1$.
5.Find the position vector of a particle at time $t$ if velocity $\vec{v}(t) = (2t\hat{i} + 3t^2\hat{j})\text{ m/s}$ and $\vec{r}(0) = \hat{i}$.
Sol: $\vec{r}(t) = \vec{r}(0) + \int_0^t (2t\hat{i} + 3t^2\hat{j}) dt = \hat{i} + t^2\hat{i} + t^3\hat{j} = (t^2 + 1)\hat{i} + t^3\hat{j}$.
