Vector and Scalar - Definition, Vector Addition and Subtraction, Differences, Solved Problems

In our day-to-day life, we come across many questions such as 'What is your height?' and 'How should a football player hit the ball to give a pass to another player on his team?" Observe that a possible answer to the first query may be 1.5 m, a quantity that involves only one value (magnitude) which is a real number. Such quantities are called scalars. However, an answer to the second query is a quantity (called force) which involves muscular strength (magnitude) and direction (in which another player is positioned). Such quantities are called vectors. In real life, we use vectors for tracking objects that are in motion, and localization of places and things.

In this article, we will cover the concept of Vectors and Scalars. This topic falls under the broader category of Vector Algebra, which is a crucial chapter in Class 11 Mathematics. This is very important not only for board exams but also for competitive exams, which even include the Joint Entrance Examination Main and other entrance exams: SRM Joint Engineering Entrance, BITSAT, WBJEE, and BCECE. A total of seven questions have been asked on this topic in JEE Main from 2013 to 2023 including two in 2021.

Physical quantities

A quantity can have magnitude or both magnitude and directions based on that quantity 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 $\vec{a}$.

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 dimensional 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 $\overrightarrow{O P}$ having O and P as its initial and terminal points, respectively, is called the position vector of the point P with respect to O.

It can also be expressed as $\overrightarrow{O P}=\vec{r}=x \hat{\mathbf{i}}+y \hat{\hat{\mathbf{i}}}$. The vectors $x \hat{\mathbf{i}}$ and $y \hat{j}$ 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 dimensional 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 $\overrightarrow{O P}=\vec{r}=x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}$

This form of any vector is called its component form. Here, x, y, and z are called the scalar components.

Sometimes x, y, and z are also called rectangular components.

Using the distance formula, the magnitude of $\overrightarrow{O P}$ or $\vec{r}$ is given by

$\overrightarrow{\mathrm{OP}} \mid=\sqrt{x^2+y^2+z^2}$

Where $\hat{\mathbf{i}}, \hat{\mathbf{j}}$ and $\hat{\mathbf{k}}$ are unit vectors parallel to the positive $X$-axis, $Y$-axis, and $Z$-axis respectively.

Vector Addition and Subtraction

Vector addition is the operation in which two vectors are added to get their sum. Since each vector may have its own direction, the process of addition of vectors is different from adding two scalars.

If a and b are two vectors, then their subtraction, or difference is called vector subtraction $\vec{a}-\vec{p}$ is defined as $\vec{a}+(-\vec{b})$, where $(-\vec{b})$ is the negative of vector b has equal magnitude but opposite direction that of b. Graphically, it is depicted by drawing a vector from the terminal point of b to the terminal point of a.

Difference Between Scalars and Vectors

The difference between Scalars and Vectors is given below:

Solved Examples Based on Vectors and Scalars

Example 1: Let O be the origin. Let $\mathrm{t} \overrightarrow{O P}=x \hat{i}+y \hat{j}-\hat{k}$ and $\overrightarrow{O Q}=-\hat{i}+2 \hat{j}+3 x \hat{k}, x, y \in R, x>0$ be such that $\overrightarrow{P Q}=\sqrt{20}$ and the vecto$\overrightarrow{Q P}$ is perpendicular to $\overrightarrow{O Q} \cdot$ If $\overrightarrow{O R}=3 i+z \hat{j}-7 \hat{k}, z c R$ is coplanar with $\overrightarrow{O P}$ and $\overrightarrow{O Q}$, then the value of $\vec{N}^2+y^2+z^2$ is equal to : [JEE MAINS 2021]

1) 9

2) 7

3) 2

4) 1

Solution

$\overrightarrow{\mathrm{OP}}=x \hat{i}+y \hat{j}-\hat{k}$ and $\overrightarrow{\mathrm{OQ}}=-\hat{i}+2 \hat{\mathrm{j}}+3 x \hat{k}$

and

$\mathrm{OP} \perp \mathrm{OQ}$
so, $-x+2 y-3 x=0 \Rightarrow y=2 x$

$\begin{aligned} & \overrightarrow{P Q}=\overrightarrow{O P}-\overrightarrow{O Q} \\ & \overrightarrow{P Q}=(\mathrm{x}+1) \hat{\mathrm{i}}+(\mathrm{y}-2) \hat{\mathrm{j}}+(3 \mathrm{x}+1) \hat{\mathrm{k}}\end{aligned}$

$\begin{aligned} & |\overrightarrow{P Q}|=\sqrt{20} \\ & (x+1)^2+(y-2)^2+(3 x+1)^2=20 \\ & (x+1)^2+(2 x-2)^2+(3 x+1)^2=20 \\ & 14 x^2+6=20 \Rightarrow x^2=1 \\ & \Rightarrow x=1, y=2\end{aligned}$

$\overrightarrow{\mathrm{OP}}, \overrightarrow{\mathrm{OQ}}, \overrightarrow{\mathrm{OR}}$ are coplanar

$
\begin{aligned}
& \Rightarrow\left|\begin{array}{ccc}
\mathrm{x} & \mathrm{y} & -1 \\
-1 & 2 & 3 \mathrm{x} \\
3 & \mathrm{z} & -7
\end{array}\right|=0 \\
& \Rightarrow\left|\begin{array}{ccc}
1 & 2 & -1 \\
-1 & 2 & 3 \\
3 & z & -7
\end{array}\right|=0
\end{aligned}
$

$\begin{aligned} & \Rightarrow 1(-14-3 z)-2(7-9)-1(-z-6)=0 \\ & \Rightarrow z=-2 \\ & \therefore x^2+y^2+z^2=1+4+4=9\end{aligned}$

Hence, the answer is option 1) 9

Example 2: A vector $\vec{d}$ has components 3p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counterclockwise sense. If, with respect to the new system, $\vec{a}$ has components $p+1$ and $\sqrt{10}$, then a value of $p$ is equal to: [JEE MAINS 2021]

|1) 1
2) -1
3) $\frac{4}{5}$
4) $-\frac{5}{4}$

Solution: Given

$\begin{aligned} & \vec{A}=3 p \hat{i}+\hat{\mathrm{j}}, \\ & \overrightarrow{A^{\prime}}=(\mathrm{p}+1) \hat{i}+\sqrt{10} \hat{j} \\ & \overrightarrow{\mathrm{A}}|=| \vec{A}^{\prime} \mid,(\text { No Change in magnitude })\end{aligned}$

$\begin{aligned} & \Rightarrow \sqrt{9 p^2+1}=\sqrt{(p+1)^2+10} \\ & 9 \mathrm{p}^2+1=\mathrm{p}^2+2 \mathrm{p}+1+10 \\ & 8 \mathrm{p}^2-2 \mathrm{p}-10=0 \\ & 4 \mathrm{p}^2-\mathrm{p}-5=0 \\ & (4 \mathrm{p}-5)(\mathrm{p}+1)=0 \\ & \mathrm{p}=-1, \mathrm{p}=\frac{5}{4}\end{aligned}$

Hence, the answer is option 2) -1

Example 3: Which of the following can't be represented using a directed line segment?

1) Force

2) Velocity

3) Acceleration

4) Mass

Solution: We know that Directed line segment - A is called the initial point and B is called the terminal point.

Mass has no direction, so it can't be represented using a directed line segment

Hence, the answer is option 4) Mass

Example 4: Which of the following are not scalars?

1) Velocity

2) Mass

3) Temperature

4) None of these

Solution: Velocity has magnitude as well as direction, but mass & temperature don't have direction

Hence, the answer is the option (1).

Example 5: Which of the following are not vectors?

1) Velocity

2) Force

3) Acceleration

4) Mass

Solution: Velocity, Force & acceleration all have magnitudes and directions, only mass doesn't have a direction so it is not a vector.

Hence, the answer is the option (4).

Summary

Scalar and vector quantity are two different aspects of determining the quantity. Scalars simplify the representation of quantities like temperature or mass, focusing on their magnitude. Meanwhile, vectors include both magnitude and direction, essential for understanding forces, velocities, and many other physical quantities. Knowledge of these concepts enhances our ability to analyze and solve real-world problems across numerous disciplines.