JEE Main Study Notes for Scalers and Vectors: Tips, Important Formulas, Previous Year's Solved Questions

JEE Main Study Notes for Scalers and Vectors have some important topics such that unit vectors, coplanar vectors, Position vector of a point, multiplication of a vector by constant, scalar products, scaler components of a vector, parallelogram law of vector, cross product, relation between direction ratios, and direction cosines and magnitude.

• The weigthage of Scalers and Vectors section is about 3.3% in JEE Main 2022.
• JEE Main Mathematics Question Paper has around 2-4 marks from the topic Scalers and Vectors section.

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What are Scalers and Vectors?

In Mathematics, Scalers and Vectors both are mathematical entities. Both Scalers and Vectors have a magnitude but only vectors have a direction. The quantities of some common scalars are distance, speed, mass, and time whereas quantities of some common vectors are force, velocity, displacement, and acceleration.

Important Factors of Vectors

Vector's magnitude defines the scalar quantity that helps to multiply the vectors

When there are the same dimensions of 2 vectors, either both of them are added or both are subtracted from each other. And the output is a vector.

For example, ‘s’ can be classified as a scalar whereas ‘v’ = (e, f) is a vector, in that position scalar multiplication can be defined as sv = s (e, f) = (se, sf). So, it can be said that each component of a vector is multiplied by the scalar.

When u, v and w are 3 vectors. On the other side, c, d are scalars, so the subsequent result of vector addition are as follows:

1) u + v = v + u

2) u + (-u) = 0

3) u + 0 = u

4) c (du) = (cd)u

5) (c + d)u = cu + d u

6) c(u + v) = cu + cv

7) 1u = u

8) u + (v + w) = (u + v) + w

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• JEE Main Study Notes for Algebra
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Representation of a Vector

Any representation of a vector has to include direction as vectors have direction. We use an arrow-headed line to represent vectors. The direction of the vector is represented by the direction of the arrow and the magnitude is represented by the length of the line. Let us start with a vector quantity called displacement. The direction of motion from point O to A is represented graphically by the arrow-headed line.

Unit Vectors

A vector having a magnitude of unity is called Unit Vector. The only purpose of a unit vector is to describe a direction in space. On a coordinate system x-y, denotes a unit vector in the positive direction x and the unit vector in the positive y-direction is 

The unit vectors and can represent any vector in the x-y plane.

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Collinear Vector

Collinear vectors are those vectors that have a common line of action. There are two types of collinear vectors -

(i) Parallel Vectors

Two or more vectors with different magnitudes are said to be parallel (θ=0°) when they are parallel to the same line. In the below image, vector and vector are parallel.

(ii) Anti Parallel Vectors

Two or more vectors that are acting in opposite directions are called anti-parallel vectors. These vectors may have different magnitudes. In the below figure, vector and vector are anti-parallel vectors.

There are more types of vectors such as Equal Vectors, Negative Vectors, Null Vectors, etc.

Check: JEE Main Probability Study Notes

Formulas of JEE Main Vectors

Some important formulas of JEE Main Vectors for JEE Main 2022 are given below.

• The magnitude of a vector can be calculated with the following formula

           [a2= b2+ c2+ d2]

• Some others formulas are l1a + m1b = l2a+ m2b

Formulas of JEE Main Scalers

Some important formulas of JEE Scalers for JEE Main 2022 are mentioned below.

• A formula for the scalar-valued function is given below.

          f(x,y,z) = x2+2yz5

Check:

• JEE Main Study notes for Integral Calculus
• JEE Main Study Notes on Differential Calculus

JEE Main Scalers and Vectors Solved Questions from Previous Year Papers

Some solved questions from the previous year’s paper regarding JEE Scalers and Vectors are given below.

Question: Rohan walks 10 m north, 12 m east, 3 m west, and 5 m south. After that, he stops to drink water. Find out the magnitude of Rohan’s displacement from the point he starts walking?

1. 10.30m
2. 10.20m
3. 11.30m
4. 10.10m

Correct Answer: (A)

Question: If a, b and c are unit vectors, what value is not exceeded through |a − b|2 + |b − c|2 + |c − a|2?

1. A) 4
2. B) 9
3. C) 8
4. D) 6

Correct Answer: (B)

Question: When three vectors which value is non-zero are a = a1i + a2j + a3k, b = b1i + b2j + b3k and c = c1i +c2j + c3k and when c is the unit vector perpendicular to the vectors a and b and also the angle between a and b is π / 6, [a1*a2*a3 is equal to

b1*b2*b3

c1*c2*c3]2

1. [1 / 4] (a12+ a22+ a23) (b21+ b22+ b23)
2. [1 / 3] (a21 + a22+ a23) (b21+ b22+ b23)
3. [1 / 2] (a21+ a22+ a23) (b21+ b22+ b23)
4. [1 / 1] (a21+ a22+ a23) (b21+ b22+ b23)

Correct Answer: ( A)

Question: When b = 4i + 3j and c be two vectors perpendicular to each other in the xy-plane. All vectors in the same plane with the projections of 1 and 2 along with b and c respectively are given by _________.

1. [−2 / 4] i + [11 / 5] j
2. [−2 / 5] i + [11 / 5] j
3. [−2 / 3] i + [11 / 5] j
4. [−2 / 1] i + [11 / 5] j

Correct Answer: (B)

Question: When the vectors ai + j + k, i + bj + k and i + j + ck (a ≠ b ≠ c ≠ 1) are coplanar, Find out the value of [1] / [1 − a] + [1] / [1 − b] + [1] / [1 − c] = _________.

1. 1
2. 2
3. 3
4. 4

Correct Answer: (A)

Question: When both b and c are any two non-collinear unit vectors whereas a is any vector, calculate the outcome of (a . b) b + (a . c) c + [a . (b × c) / |b × c|] (b × c) = ___________.

1. a
2. b
3. c
4. d

Correct Answer: (A)

Question: When p, q, r are three mutually perpendicular vectors of the same magnitude, When a vector x satisfies equation p × {(x − q) × p} + q × {(x − r) × q} + r × {(x − p) × r} = 0, then calculate the value of x.

1. [1 / 2] (p + q+ r)
2. [1 / 2] (p + q+ r)
3. [1 / 2] (p + q+ r)
4. [1 / 2] (p + q+ r)

Correct Answer: (A)

Question: When a = 2i + j − 2k and b = i + j. If c is taken as a vector so that a . c = |c|, |c − a| = 2√2 whereas the angle between (a × b) and c is 30o, so, |(a × b) × c| = _________.

1. 3/2
2. 3/6
3. 3/5
4. 3/1

Correct Answer: (A)

Question: When a, b, c are different non-negative numbers and when the vectors ai + aj + ck, i + k and ci + cj + bk are placed in a plane, what is the value of c?

1. A) The arithmetic mean of a and b
2. B) The geometric mean of a and b
3. C) The harmonic mean of a and b
4. D) Equal to zero

Correct Answer: (B)

Question: a. [(b + c) × (a + b + c)] is equal to ______.

1. 0
2. 1
3. 2
4. 3

Correct Answer: ( A)

Question: The horizontal force and the force inclined at an angle of 60° with the vertical, whose resultant is in the vertical direction of ‘P’ kg, are ________.

1. 2P
2. 3P
3. 4P
4. 6P

Correct Answer: ( A)

JEE Main Scalers and Vectors Preparation Tips

Some tips for the preparation of the section JEE Main Scalers and Vectors are listed below.

• Give more importance to the accuracy level. By solving questions from different study material, previous years’ papers, one can increase their accuracy level
• Always try to decrease the solution time. It will help to accelerate your speed to complete all the questions on time in the exam
• Do practice all the important formulas and try to learn and practice shortcuts. It will help to save your time in the exam.
• There are multiple books that you will find for this section of JEE Main 2022 but you need to select some recommended books for better preparation.

Check JEE Main Mathematics Preparation Books

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