Concept of vectors for IIT JEE!

Willard Gibbs was basically a mathematical physicist whose contribution in statistical

mechanics paved the way for physical

chemistry as a science. Einstein regarded

Gibbs as the “greatest mind in American

history.” Gibbs is also popularly known as the

father of vector analysis or the proper study

of vectors in math. The topic of vectors is an

extremely important topic of IIT JEE

Mathematics syllabus. The topic is not very

tough and can be easily mastered with a bit of

practice. Here, we shall discuss vectors in detail and throw some light on various

interesting facts including its sub-topics. Besides, we shall also illustrate some

important tips to master the topic in addition to its practical applications in real life.

Various sub-topics of vectors include:

Introduction to vectors

Basic Vector Operations

Non-Coplanar Vectors

Resolution of a Vector

Dot Product

Cross Product

Scalar Triple Product

Vector Triple Product

Geometrical Interpretations

Those willing to study these topics in detail are advised to refer the study material

section of vectors.

Vectors

All measurable quantities are termed as physical

quantities and the physical quantities can be

categorized as: The quantities which can be described

only by specifying their magnitude are termed as

scalars. Some common examples include: mass, work,

distance, speed etc. Those physical quantities that

require both magnitude and direction to describe them

completely are called as vectors. Some examples of

vector quantities include velocity, displacement,

acceleration, momentum and force. A vector can hence be assumed to be a directed

line segment which has a certain magnitude and points in a certain direction.

Some Interesting Results and Formulae

The magnitude of a vector is a scalar and scalars are denoted by normal letters.

Vertical bars enclosing a boldface letter denote the magnitude of a vector. Since

the magnitude is a scalar, it can also be denoted by a normal letter, |w| = w denotes

the magnitude of a vector.

The vectors are denoted by either drawing an arrow above the letters or by

boldfaced letters.

Vectors can be multiplied by a scalar and the result is again a vector.

A vector divided by its magnitude is a unit vector, i.e. =

Two vectors and are said to be equal if they have the same magnitude and the

same direction.

Suppose c is a scalar and = (a, b) is a vector, then the scalar multiplication is

defined by c = c (a,b)= (ca, cb). Hence each component of a vector is multiplied

by the scalar.

If two vectors are of the same dimension then they can be added or subtracted

from each other. The result is again a vector. In such a case, the sum of these

vectors is defined by + = (a + e, b + f), where = (a, b) and = (e, f).

We can also subtract two vectors of the same direction. The result is again a

vector. As in the previous case subtracting vector from yields - = (a -

e, b – f). The difference of these vectors is actually the vector - = +(-1)

Dot Product of Two Vectors a and b where a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is

given by a1b1 + a2b2 + ..., + anbn .

Some basic rules

If , and are three vectors and a, b are scalars then the following results hold

true:

Commutative law of addition: + = +

Existence of additive inverse: +(- ) =

+ =

a(b ) = (ab)

(a+b) = a + b

a( + ) = a + a

1. =

Associative law of addition: +( + ) = ( + )+

Triangle Law of Vector Addition:

Given two vectors and , their sum or resultant written as ( + ) is also a

vector obtained by first bringing the initial point of b to the terminal point of a and

then joining the initial point of a to the terminal point of b giving a uniform direction

by completing the triangle OAB.

Parallelogram Law of Vector Addition

If two adjacent sides of a parallelogram are represented by two vectors, then the

diagonal of the parallelogram is represented by the resultant of both the vectors.

Section Formula

If there is a line segment AB with coordinates A(x1, y1) and B (x2, y2) and P is a point

that divides it in the ratio m:n, then by Section Formula we have ((m1x2 + m2x1)/(m1 +

m2), (m1y2 + m2y1)/(m1 + m2))

What is the importance of this topic for an IIT JEE aspirant?

Vectors occupy an extremely important place in IIT JEE preparation. It accounts for

5% of the JEE screening. For majority of students, vector is the most interesting

topic encountered by them in JEE Mathematics. The best part is that it also fetches

good number of questions in JEE.

What are the Best Books for the Preparation of Vectors?

Students are advised not to miss the NCERTs as they help in laying good foundation.

Besides NCERTS, several other books which are considered to be excellent for JEE

preparations, especially for vectors include:

Arihant Publication

R.D.Sharma

IIT Maths by M.L. Khanna

These books not only explain the concepts in detail but also contain various solved and

unsolved examples. Various JEE level questions are also included in them. Arihant

Calculus is excellent in every respect except that students need to select the

problems carefully as the ones designed for Olympiads are not very useful for JEE

aspirants. R.D.Sharma offers plenty of solved examples which help in clearing the

concepts of students.

Illustrations

Illustration 1:

If =( + + ), . = 1 and x = - , then is (IIT JEE 2003)

A. - + C.

B. - D. 2

Solution: We know that x ( x ) = ( . ) – ( . )

Hence, ( + + ) x ( - )= ( + + ) – (√3)2

This gives, -2 + + = + + – 3

Hence, 3 = 3

And this yields =

Illustration 2:

If the vector = + + , the vector = + + and = + α + β are

linearly dependent vectors and | | = √3 then,

A. α = 1, β = -1 B. α = 1, β = ±1

C. α = -1, β = ±1 D. α = ±1, β = 1

Solution: Since , and are given to be linearly dependent vectors, so this also

implies that the vectors are coplanar.

So, . ( x ) = 0

Solving this with the help of determinants, and applying the column operation

C3 – C1, we get,(β-1) (3-4) = 0.This gives β = 1.Also, | | = √3

√ 1+ α2 + β2 = √3, √ 1+ α2 +1 = √3

√α2 + 2 = √3, α2 + 2 = 3 so α = ±1.

Some Interesting Facts

Vectors which are in the same direction can be added by simply adding their

magnitudes, while to add two vectors pointing in opposite directions their

magnitudes are subtracted and not added.

Column vectors can be added by simply adding the values in each row.

You can find the magnitude of a vector in three dimensions by using the

formula a2=b2+c2+d2, where a is the magnitude of the vector, and b, c, and d are the

components in each direction.

If l1 + m1 = l2 + m2 then l1 = l2 and m1 = m2

Cross product of vectors is not commutative.

Collinear Vectors are also parallel vectors except that they lie on the same line.

Three vectors are linearly dependent if they are coplanar which implies that any

one of them can be represented as a linear combination of other two.

The dot product of two parallel vectors is 1 and their cross product is 0.

Two collinear vectors are always linearly dependent but two non-collinear non-zero

vectors are always linearly independent.

Three coplanar vectors are always linearly dependent.

Three non-coplanar non-zero vectors are always linearly independent.

More than 3 vectors are always linearly dependent.

Minimum number of coplanar vectors for zero resultant is 2(for equal magnitude)

and 3 (for unequal magnitude).

DCs of a vector are unique and satisfy the relation l2 + m2 + n2 = 1, where l, m and n

are the three DCs.

DRs are proportional to DCs and are not unique.

Tips to Master Vectors for IIT JEE

Vector is a simple topic but it demands conceptual clarity and

consistent practice. Due to lack of conceptual clarity, students

often end up committing silly mistakes which results in loss of

some easy scores. Listed below are some of the tips which can help you fetch perfect

scores in this simple topic:

Consulting too many books can lead to confusion and hence students must refer

just one or two books after they are done with the NCERTs.

Vectors are more of a geometrical concept rather than algebraic. Hence, when we

talk of (say) addition of two vectors, rather than assuming it to be simple addition

students are advised to visualize the concepts.

Vectors are free entities and hence one can move around a vector in the plane and

it will remain the same vector as long as its direction and magnitude are preserved.

Consistent practice is a must to excel in this topic.

Various results and formulae should be on fingertips.

While attempting a question on vectors, students are advised to visualize the

concept and draw a diagram for the same so that chances of committing mistakes

are minimized.

The concepts of dot product and cross product including their nature must be very

clear.

Suppose we have three non-zero vectors , & , then if their scalar triple

product is zero, then the three vectors are coplanar.

Practice sufficient number of previous year papers to get acquainted with the kind

of questions asked in exam.

Do not read the questions; try to depict them by figures to reach at the solution

easily.

Practical Applications of Vectors for IIT JEE

Vector is a comparatively new topic in the history of

mathematics but it has gained immense popularity in this short

span of time. It has wide range of applications in solving

geometric problems especially in the area of computer graphics. Some of the practical

applications include:

The concepts of vector dot and cross products are used to estimate the capability

of solar panels to produce the electrical power.

The concept of unit vectors is used in determining the directions like eats, north

etc. Whenever and wherever direction is to be determined in real life, unit vectors

play a vital role.

Flying planes are based on the concept of vector mathematics. The motion of the

plane is along the vector sum of wind and propeller velocity.

Driving and sailing also use the vector principles, although we don’t really realize it.

If one needs to cross a flowing river, the concept of vectors is used in order to

determine the point where he will land on the side.

A bicycler tilts his umbrella forward while riding on a bicycle because the relative

velocity of the rain is slant and hits his face. The more the speed of the cycle, the

more he should tilt forward.

Controlling and tracing the movement of cars, ships and planes moving in space

involve the use of unit vectors.

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