1.Define a unit vector. What is its magnitude and physical purpose?
Sol: A unit vector has magnitude $1$ and specifies direction. $\hat{A} = \frac{\vec{A}}{|\vec{A}|}$.
2.State the triangle law of vector addition.
Sol: If two vectors are represented in magnitude and direction by two sides of a triangle taken in order, their resultant is given by the third side taken in reverse order.
3.What is the dot product of two perpendicular vectors?
Sol: Zero, because $\cos 90^\circ = 0$.
4.Calculate the magnitude of $\vec{A} = 3\hat{i} + 4\hat{j} + 12\hat{k}$.
Sol: $|\vec{A}| = \sqrt{3^2 + 4^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13$.
5.Find the angle between $\vec{A} = \hat{i} + \hat{j}$ and $\vec{B} = \hat{j} + \hat{k}$.
Sol: $\vec{A} \cdot \vec{B} = 1(0) + 1(1) + 0(1) = 1$. $|\vec{A}| = \sqrt{2}, |\vec{B}| = \sqrt{2}$. $\cos \theta = \frac{1}{2} \implies \theta = 60^\circ$.
6.Find the cross product $\vec{A} \times \vec{B}$ for $\vec{A} = 2\hat{i} + 3\hat{j}$ and $\vec{B} = \hat{i} + 4\hat{j}$.
Sol: $\vec{A} \times \vec{B} = (2 \cdot 4 - 3 \cdot 1)\hat{k} = 5\hat{k}$.
7.A force $\vec{F} = (5\hat{i} + 3\hat{j} + 2\hat{k})\text{ N}$ displaces a body by $\vec{d} = (2\hat{i} - \hat{j})\text{ m}$. Find work done.
Sol: $W = \vec{F} \cdot \vec{d} = 5(2) + 3(-1) + 2(0) = 10 - 3 = 7 \text{ Joules}$.
8.Find the projection of $\vec{A} = 2\hat{i} + 3\hat{j}$ on $\vec{B} = \hat{i} + \hat{j}$.
Sol: Projection = $\frac{\vec{A} \cdot \vec{B}}{|\vec{B}|} = \frac{2(1) + 3(1)}{\sqrt{1^2 + 1^2}} = \frac{5}{\sqrt{2}}$.
