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.
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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.
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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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.
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 vectors are those vectors that have a common line of action. There are two types of collinear 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.
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
Some important formulas of JEE Main Vectors for JEE Main 2022 are given below.
[a2= b2+ c2+ d2]
Some important formulas of JEE Scalers for JEE Main 2022 are mentioned below.
f(x,y,z) = x2+2yz5
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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?
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?
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
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 _________.
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] = _________.
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) = ___________.
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.
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| = _________.
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?
Correct Answer: (B)
Question: a. [(b + c) × (a + b + c)] is equal to ______.
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 ________.
Correct Answer: ( A)
Some tips for the preparation of the section JEE Main Scalers and Vectors are listed below.
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*The article might have information for the previous academic years, please refer the official website of the exam.