Vector algebra is one of the fundamental topics of algebra. There are two types of physical quantities – scalar quantity and vector quantity. In vector algebra, we study the vector quantities. Vector quantity is a quantity that has both magnitude and direction and follows vector addition rules whereas scalar quantity only has magnitude.

This Story also Contains

Vectors Addition Rules
Subtraction And Addition Of Vectors
Multiplication Of Vectors
Other Important Formulas
Logarithmic
Exponential
Exponential Function Rules
NEET Previous Five Years’ Analysis
Previous Years Questions To Understand The Concepts

NEET Five-Year Analysis: Solving Vectors Algebra, Logarithmic, And Exponential Questions

NEET Vector Algebra(Image: Shutterstock)

Vector: A vector is a measurement or quantity that includes both magnitude and direction. Generally, it can be represented physically by an arrow, and mathematically as A⃗ can be read as “A vector”. Arrow (➙) indicates the direction of the vector and its length represents the magnitude of the vector that is given in the following figure. |A⃗| represents the magnitude of the vector A. If we say that two vectors are equal, then both must have the same direction as well as magnitude.

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Unit Vector: It is a vector that has a unit magnitude. Generally, it is used to show the direction of the vector. Conventionally three-unit vectors namely $\hat{i}, \hat{j}, and \ \hat{k}$ are used in the direction of the x-axis, y-axis, and z-axis.

The unit vector can be calculated by dividing the vector by its magnitude and represented as mentioned below.

$\hat{a} = \frac{\vec{a}}{|\vec{a}|}$

Vectors Addition Rules

Triangle Rule

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Let us consider two vectors and represented by two sides of a triangle, the head of is connected with the tail of as shown above in the image, then the third side in the opposite direction represents the resultant sum of these two vectors. Mathematically the resultant of two vectors can be calculated by the given formula.

$|\vec{A} + \vec{B}| = \sqrt{A^2+B^2+2ABcos \theta}$

$tan \alpha = \frac{Bsin\theta}{A+Bcos\theta}$

Where angle ⍺ is the angle between the resultant R vector and . ? is the angle between and . Remember that angle between two vectors can only be calculated when both vectors are connected head to head or tail to tail.

Parallelogram Rules

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Let's consider two adjacent sides of a parallelogram represent two vectors and . As mentioned above in the diagram, the diagonal joining the intersecting point to the other corner of the parallelogram represents the resultant sum of vectors as shown in the image.

Subtraction And Addition Of Vectors

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Subtraction of the vector is nothing but the addition of the negative vector to the other vector. Keeping the same magnitude and reversing the direction of a vector gives a negative vector. In a Parallelogram, one diagonal shows the addition of vectors, and the other one shows the subtraction of vectors as shown above in the figure.

Mathematical formula of subtraction of two vectors and

$|\vec{A} - \vec{B}| = \sqrt{A^2+B^2-2ABcos \theta}$

$tan \alpha = \frac{Bsin\theta}{A-Bcos\theta}$

Where ? is the angle between and and angle ? is between resultant vector $\vec{A} - \vec{B}$ and

Multiplication Of Vectors

Multiplication of vectors can be of two types: scalar vector multiplication and vector-vector multiplication. If we multiply a vector with a scalar quantity “k”, when “k” is positive, the direction remains the same and magnitude becomes “k” times, and when “k” is negative, magnitude becomes “k” time, but the direction gets reversed.

Vector multiplication is two types i.e dot product or scalar product and cross product or vector product which are explained below.

Dot Product

It is also called a scalar product, represented as (.) between two vectors. Dot product physically represents the multiplication of the magnitude of one vector with a component of another vector in the direction of the vector.

$\\ \vec{P}.\vec{Q} = |\vec{P}| |\vec{Q}| cos \theta \\ \text{If P vector and Q vector are in the same direction, i.e. } \theta = 0 {\degree} \\ then; \vec{P}.\vec{Q} = |\vec{P}| |\vec{Q}| \\ \text{If P and Q are both orthogonal, i.e. } \theta = 90 \degree , \\ then; \vec{P}.\vec{Q} = \vec{0} \ \text{since } cos 90 \degree = 0 \\$

$\\ \vec{A} = a_{1} \ \hat{i}+a_{2} \ \hat{j}+a_{3} \ \hat{k} \\ \vec{B} = b_{1} \ \hat{i}+b_{2} \ \hat{j}+b_{3} \ \hat{k} \\ \vec{A}.\vec{B} = a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}$

Cross Product

Cross product is also known as the vector product is represented by the “X” sign between two vectors. If and are two vectors.

$\\ \vec{A} = a_{1} \ \hat{i}+a_{2} \ \hat{j}+a_{3} \ \hat{k} \\ \vec{B} = b_{1} \ \hat{i}+b_{2} \ \hat{j}+b_{3} \ \hat{k} \\ \vec{A} \times \vec{B} = \begin{vmatrix} i & j & k\\ a_{1} & a_{2} & a_{3}\\ b_{1}& b_{2} & b_{3} \end{vmatrix}$

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Other Important Formulas

$\hat{a} = \frac{\vec{a}}{|\vec{a}|}$

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Logarithmic

Logarithmic Function is an inverse function of exponential function which is defined as

x = log_ba, for b > 0, a > 0, and b ≠ 1, if and only if a = b^x. Generally, we come across two types of bases while solving NEET problems. That is 10 and the mathematical constant “e”. When the base is “e” it is called a natural logarithmic function and when the base is 10 then it is called a common logarithmic function.

Natural logarithmic function f(x) = log_e x

Common logarithmic function f(x) = log₁₀ x

Logarithmic Function

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Logarithmic Table

x	Base = 10 Log₁₀ x	Base = e ≈ 2.71; Log_e x
1	0	0
2	0.3010	0.693147
3	0.4771	1.098612
4	0.6020	1.386294
5	0.6989	1.609438
6	0.7781	1.791759
7	0.8450	1.94591
8	0.9030	2.079442
9	0.9542	2.197225
10	1	2.302585

Graph

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Properties

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Exponential

It is a mathematical function that is used to find exponential decay and exponential growth of many real-life events like population models, radioactive decay, wounds healing models, and many more. Mathematically defined by a function f (x) = a.b^x, where “x” is a variable that can be any real number, “a” is constant but can not be zero, and “b” is an also constant know as base of exponential function and it should be greater than zero and not equal to 1. And the most commonly used exponential bases are 10 and “e” which is approximately equal to 2.718. This article includes rules, formulas, graphs, etc of the exponential function.

Exponential Growth

Initially, the quantity increases very slowly, and then rapidly. The rate of change increases over time. A mathematical expression that defines exponential growth is y = a ( 1+ r )^x where “ r ” is the growth percentage.

Exponential Decay

Initially, quantity decreases very rapidly, and then slowly. The mathematical expression to define exponential decay is y = a ( 1- r )^x where “ r ” is the decay percentage.

Exponential Growth And Decay Graph

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Exponential And Logarithmic Function

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Exponential Function Rules

Following are some important rules that are repeatedly used. Here x and y are real numbers whereas “a” and “b” are positive numbers.

If a>0, and b>0, the following hold true for all the real numbers x and y:

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NEET Previous Five Years’ Analysis

Since the NEET exam is highly competitive, smart study can help students to succeed in the exam. Careers360 came up with an analysis to help them. The following list shows the number of questions in which vector algebra, exponential, and logarithmic concepts are used.

Numbers Of Questions Asked In Previous Five Years NEET Exam

	Vector Algebra	Exponential	Logarithmic
2021	4	2	1
2020	1	2	2
2019	5	2	1
2018	4	1	1
2017	1	3	1

Previous Years Questions To Understand The Concepts

Here is a list of previous year's questions with step-by-step explanations to understand vector algebra, exponential, and logarithmic concepts. NEET aspirants can develop an understanding of the level of questions along with the spread and depth of the concepts.

Q.1 NEET - 2021

In the product

What will be the complete expression for

Solution:

$\bar{F}=q\left ( \bar{v}\times \bar{B} \right )=q\bar{v}\times \left ( B\hat{i}+B\hat{j}+B_{0}\hat{k} \right )$ $\left [ 4\hat{i}-20\hat{j}+12\hat{k} \right ]=\left [ 2\hat{i}+4\hat{j}+6\hat{k} \right ]\times \left [ B\hat{i}+B\hat{j}+B_{0}\hat{k} \right ]$

$=\hat{i}\left ( 4B_{0}-6B \right )-\hat{j}\left ( 2B_{0}-6B \right )$ $+\hat{k}\left ( -2B \right )$

$4B_{0}-6B=4$

$2B_{0}-6B=20$

$2B_{0}=-16,B_{0}=-8$

$\bar{B}=B\hat{i}+B\hat{j}+B_{0}\hat{K}$

$\bar{B}=-6\hat{i}-6\hat{j}-8\hat{k}$

Concepts used:

Cross product formula:

Q.2. NEET - 2020

Find the torque about the origin when a force of $3\hat j N$ acts on a particle whose position vector is $2\hat k m$

Solution:

$\\ \vec{F}=3 \hat{j}\\ \vec{r}=2 \hat{k}\\ \tau =\vec{r}\times \vec{F}=2 \hat{k}\times 3 \hat{j}=-6 \hat{i}$

Concepts used:

Cross product formula:

Q.3. NEET - 2021

If force [F], acceleration [A], and time [T] are chosen as the fundamental physical quantities. find the dimensions of energy.

Solution:

$[Energy ]=[F]^{x}[A]^{y}[T]^{Z}$

$\left[M^{1} L^{2} T^{-2}\right]=\left[M^{1} L^{1} T^{-2}\right]^{x}[L^{1}T^{-2}]^{y}[T^{1}]^{z}$

$\left[M^{1} L^{2} T^{-2}\right]=M^{x} L^{x+y} T^{-2 y-2 x+z}$

Concepts Used:

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Q.4. NEET - 2020

The rate constant for a first-order reaction is $4.606 \times 10^{-3}s^{-1}$ The time required to reduce 2 g of the reactant to 0.2 g is:

Solution:

We have given $K=4.606 \times 10^{-3}s^{-1}$ Thus, the reaction is of the first order. $\mathrm{K\: t\: =\: ln\frac{A_{o}}{A}}$

$\mathrm{4.606\: x\: 10^{-3}\: t\: =\: 2.303log\left (\frac{2}{0.2} \right )}$

Thus,

$\text {t}=500\; \text {sec}$

Concepts Used:

Log₁₀ 10 = 1

Q.5 NEET - 2020

The slope of the Arrhenius Plot $\left(\ln \mathrm{k\ v} / \mathrm{s}\ \frac{1}{\mathrm{T}}\right)$ of a first-order reaction is $-5 \times 10^{3} K$ . The value of of the reaction is. Choose the correct option for your answer.

Solution:

We know the Arrhenius equation

$\mathrm{k}=\mathrm{Ae}^{-\mathrm{E}_{\mathrm{a}} / \mathrm{RT}}$

$\ln \mathrm{k}=\ln \mathrm{A}+\ln \mathrm{e}^{-\mathrm{E}_{\mathrm{a}} / \mathrm{RT}}$

$\mathrm{\ln k=\ln A-\frac{E_{a}}{R}\left(\frac{1}{T}\right)}$

The slope of $\left(\ln \mathrm{k\ v} / \mathrm{s}\ \frac{1}{\mathrm{T}}\right)$ of the above equation -

$\mathrm{m=-\frac{E_{a}}{R}}$

Given, m = $-5 \times 10^{3} \mathrm{~K}$ and $\mathrm{R}=8.314 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}$

$\mathrm{-5 \times 10^{3}=-\frac{E_{a}}{8.314}}$

$\mathrm{E_{a}=5 \times 10^{3} \times 8.314 \mathrm{~J} / \mathrm{mol}}$

$\mathrm{E_{a} =41.57 \times 10^{3} \mathrm{~J} / \mathrm{mol}}$

$\mathrm{E_{a}\simeq 41.5 \mathrm{~kJ} / \mathrm{mol}}$

Concepts Used:

Using property ln x^p = p ln x, simplify the equation and ln e = 1.

After going through this article it is clear that the NEET exam does not go deeper and asks just formula-based and simple concept-based questions. Therefore, even without mastering complex formulas and concepts of vector algebra students can score well in the exam. With continuous practice, aspirants can get command of the concepts.

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Neet pervious 5 years question paper with solution pdf(2021-2025) free download

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Maharashtra MBBS/BDS Counselling

GSMC Mumbai - Seth GS Medical College, Mumbai

LTMMC Mumbai - Lokmanya Tilak Municipal Medical College, Sion, Mumbai

Arunachal Pradesh MBBS/BDS Admissions

KGMU Lucknow - King George's Medical University, Lucknow

Dr R Ahmed Dental College and Hospital, Kolkata

NEET UG

BMCRI Bangalore - Bangalore Medical College and Research Institute, Bangalore

KMC Mangalore - Kasturba Medical College, Mangalore

Karnataka MBBS/BDS Counselling

BMCRI Bangalore - Bangalore Medical College and Research Institute, Bangalore

KMC Mangalore - Kasturba Medical College, Mangalore

UP MBBS/BDS Counselling

KGMU Lucknow - King George's Medical University, Lucknow

Dr Ram Manohar Lohia Institute of Medical Sciences, Lucknow

Gujarat MBBS/BDS Counselling

BJMC Ahmedabad - B J Medical College, Ahmedabad

GMERS Medical College and Hospital, Sola

Madhya Pradesh MBBS/BDS Counselling

Gandhi Medical College, Bhopal

Gajra Raja Medical College, Gwalior

Kerala MBBS/BDS Counselling

Government Medical College, Kozhikode

Government Medical College, Thiruvananthapuram

Tamil Nadu MBBS/BDS Counselling

MMC Chennai - Madras Medical College, Chennai

Stanley Medical College, Chennai

Telangana MBBS/BDS Counselling

Osmania Medical College, Hyderabad

ESIC Medical College, Hyderabad