Vectors and Scalars - Practice Questions & MCQ

Physical quantities are divided into two categories- Scalar quantities and Vector quantities

Scalar Quantity

A quantity that has magnitude but no direction is called a scalar quantity (or scalar), e.g., mass, volume, density, speed, etc. A scalar quantity is represented by a real number along with a suitable unit.

Vector Quantity

A quantity that has magnitude as well as a direction in space and follows the triangle law of addition is called a vector quantity, e.g., velocity, force, displacement, etc.

In this text, we denote vectors by boldface letters, such as a  or .

Representation of a Vector

A vector is represented by a directed line segment (an arrow). The endpoints of the segment are called the initial point and the terminal point of the vector. An arrow from the initial point to the terminal point indicates the direction of the vector. 

The length of the line segment represents its magnitude. In the above figure, a = AB, and the magnitude (or modulus) of vector a is denoted as $|\vec{a}|=|\overrightarrow{A B}|=A B$ (Distance between the Initial and terminal point).

The arrow indicates the direction of the vector.

Position Vector

In Two dimension system

Let P be any point in the x-y plane, having coordinates (x, y) with respect to the origin O(0, 0, 0).

Then, the vector  having O and P as its initial and terminal points, respectively, is called the position vector of the point P with respect to O.

It can also be expressed as $\overrightarrow{O P}=\vec{r}=x \hat{\mathbf{i}}+y \hat{\mathbf{j}}$. The vectors are called the perpendicular components of vector r. Where  and are unit vectors (vectors of length equal to 1) parallel to the positive X-axis and positive Y-axis respectively.

The magnitude of $\tilde{\mathbf{r}}=\sqrt{\mathrm{x}^2+\mathrm{y}^2}$ and if $\theta$ is the inclination of $\tilde{\mathbf{r}}$ w.r.t. $\mathrm{X}-$ axis, then, $\theta=\tan ^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)$.

In Three dimension system

Let $P$ be any point in space, having coordinates $(x, y, z)$ with respect to the origin $O(0,0,0)$.
Then, the vector $\overrightarrow{O P}$ having O and P as its initial and terminal points, respectively, is called the position vector of the point P with respect to O .
OP vector can also be expressed as $\overrightarrow{O P}=\vec{r}=x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}$
Using distance formula, the magnitude of $\overrightarrow{O P}$ or $\vec{r}$ is given by
$$
|\overrightarrow{\mathrm{OP}}|=\sqrt{x^2+y^2+z^2}
$$

 Where , and are unit vectors parallel to the positive X-axis, Y-axis, and Z-axis respectively.