This page contains Vector questions for B.Sc. first year mechanics. You can also practice these questions if you are appearing for any after B.Sc. entrance exams(ex. IIT JAM) for pursuing M.Sc.

**Question 1**

Define direction cosines of a vector. If $l_1$, $m_1$, $n_1$ and $l_2$, $m_2$, $n_2$ are the direction cosines of two vectors show that the angle $\theta$ between them is given by $cos\theta = l_1l_2 +m_1m_2 + n_1n_2$.

**Question 2**

If vector $\vec A = \hat i + 2\hat j – 2\hat k$ . $\vec B = 2\hat i + \hat j + \hat k$ and $\vec C = \hat i – 3\hat j – 2\hat k$ find the magnitude and direction cosines of the vector ($\vec A+\vec B+\vec C$).

**Question 3**

Define scalar (dot) product of two vectors.

**Question 4**

Prove that $\hat A = \hat icos(\vec A,x) + \hat jcos(\vec A,y) + \hat kcos(\vec A,z)$ where $\hat A$ is a unit vector and $cos(\vec A,x)$ , $cos(\vec A,y)$ , $cos(\vec A,z)$ are direction cosines of $\vec A$.

**Question 5**

(a) Deduce the expression for the scalar product of two vectors in terms of their rectangular components.

(b) Hence derive an expression for the angle between two vectors.

**Question 6**

If $\vec A=\lambda \vec B$ Prove that

$$\frac{A_x}{B_x}= \frac{A_y}{B_y}= \frac{A_z}{B_z}$$

**Question 7**

If $|\vec A|=|\vec B|$ prove that $(\vec A + \vec B)$ is perpendicular to $(\vec A – \vec B)$

**Question 8**

Define cross-product of two vectors.

**Question 9**

(a) Deduce the expression for vector product of two vectors in terms of their components.

(b) Hence derive an expression for the angle between two vectors.

**Question 10**

(a) Define torque. Show that torque can be written as

$\vec \tau = \vec r \times \vec F$ .

(b) Derive a relation between angular momentum of a particle, its mass and angular velocity. Which property of vector is used in deriving this relation?