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Physics · Class 11 Science

Ch 2Mathematical Methods — Class 11 Physics, concept-first.

Physics describes the physical world using precise mathematical language, and this chapter introduces two mathematical tools that you will use throughout your study of physics: vector analysis and an elementary introduction to differential and integral calculus.

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Vector Addition and Subtraction

Only vectors describing the SAME physical quantity may be combined by addition or subtraction — two forces can be added, but a force cannot be added to a velocity.

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Chapter contents

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2.1

Introduction

Physics describes the physical world using precise mathematical language, and this chapter introduces two mathematical tools that you will use throughout your study of physics: vector analysis and an…

2.2

Vector Analysis

Not every physical quantity can be completely specified by a number and a unit alone. Consider a man who walks for a certain time at a certain speed: knowing only the time and the speed tells you the…

2.2.1

Scalars

A physical quantity that can be completely described by its magnitude alone is called a scalar. Stating a scalar quantity means giving a number together with the appropriate unit; the two together giv…

2.2.2

Vectors

A physical quantity that needs both a magnitude and a direction for its complete description is called a vector. Displacement, velocity, and force are common examples.

2.3

Vector Operations

Having defined what a vector is and the different kinds of vectors, the next step is to learn the operations that can be performed with vectors: multiplying a vector by an ordinary number (scalar), an…

2.3.1

Multiplication of a Vector by a Scalar

Multiplying a vector by an ordinary number (a scalar) produces another vector, The resulting vector has the same direction as (for positive ) and a magnitude equal to times the magnitude of , i.e. .

2.3.2

Addition and Subtraction of Vectors

Only vectors describing the SAME physical quantity can be added or subtracted — for instance, two forces and can be combined into a resultant force , but a force vector can never be added to a velocit…

2.3.3

Triangle Law for Vector Addition

When two vectors describing the same physical quantity do not act along the same or opposite directions, their resultant can be found using the triangle law of vector addition, stated as follows: if t…

2.3.4

Law of parallelogram of vectors

Another geometric method for adding two vectors is the parallelogram law of vector addition, stated as follows: if two vectors of the same type, with their tails at the same point, are represented in…

2.4

Resolution of vectors

Just as two or more vectors can be combined into a single resultant vector, a single vector can conversely be expressed as the sum of two or more vectors along chosen fixed directions: where are unit…

2.5

Multiplication of Vectors

Vectors of the same type can be added or subtracted to produce a resultant of the same type. Multiplication of vectors is different: multiplying two vectors together produces a NEW physical quantity,…

2.5.1

Scalar Product (Dot Product)

The scalar product (or dot product) of two non-zero vectors and is defined as the product of their magnitudes and the cosine of the angle between them: The result is a SCALAR (a pure number with units…

2.5.2

Vector Product (cross product)

The vector product (or cross product) of two vectors and is a VECTOR whose magnitude equals the product of their magnitudes and the sine of the smaller angle between them, and whose direction is perpe…

2.6

Introduction to Calculus

Calculus is the branch of mathematics that studies continuous (as opposed to discrete, step-by-step) change in mathematical quantities.

2.6.1

Differential Calculus

Consider a function , where is the independent variable and is the dependent variable — for example, could be the position of a particle and its velocity at that position.

2.6.2

Integral calculus

Integral calculus deals with the properties and applications of integrals. Physically, the integral of a function , written , represents the AREA under the curve of plotted against .

More questions

17 Q
+Show 5 questions5 questions
  1. Q1The resultant of two forces 10 N and 15 N acting along + x and − x-axes respectively, is (A) 25 N along + x-axis (B) 25 N along − x-axis (C)…Free
  2. Q2For two vectors to be equal, they should have the (A) same magnitude (B) same direction (C) same magnitude and direction (D) same magnitude…Free
  3. Q3The magnitude of scalar product of two unit vectors perpendicular to each other is (A) zero (B) 1 (C) −1 (D) 2Preview
  4. Q4The magnitude of vector product of two unit vectors making an angle of 60° with each other is (A) 1 (B) 2 (C) 3/2 (D) √3/2Preview
  5. Q5If $\vec{A}$, $\vec{B}$ and $\vec{C}$ are three vectors, then which of the following is not correct? (A) $\vec{A}\cdot(\vec{B}+\vec{C}) = \v…Preview
+Show 5 questions5 questions
  1. Q6Show that $\vec{a} = \dfrac{\hat{i}-\hat{j}}{\sqrt{2}}$ is a unit vector.Free
  2. Q7If $\vec{v_1} = 3\hat{i}+4\hat{j}+\hat{k}$ and $\vec{v_2} = \hat{i}-\hat{j}-\hat{k}$, determine the magnitude of $\vec{v_1}+\vec{v_2}$.Free
  3. Q8For $\vec{v_1} = 2\hat{i}-3\hat{j}$ and $\vec{v_2} = 6\hat{i}+5\hat{j}$, determine the magnitude and direction of $\vec{v_1}+\vec{v_2}$.Preview
  4. Q9Find a vector which is parallel to $\vec{v} = \hat{i}-2\hat{j}$ and has a magnitude 10.Preview
  5. Q10Show that vectors $\vec{a} = \dfrac{2}{5}\hat{i}+\hat{j}-\dfrac{6}{5}\hat{k}$ and $\vec{b} = \hat{i}+\dfrac{5}{2}\hat{j}-3\hat{k}$ are paral…Preview
+Show 7 questions7 questions
  1. Q11Determine $\vec{a}\times\vec{b}$, given $\vec{a} = 2\hat{i}+3\hat{j}$ and $\vec{b} = 3\hat{i}+5\hat{j}$.Free
  2. Q12Show that vectors $\vec{a} = 2\hat{i}+3\hat{j}+6\hat{k}$, $\vec{b} = 3\hat{i}-6\hat{j}+2\hat{k}$ and $\vec{c} = 6\hat{i}+2\hat{j}-3\hat{k}$…Free
  3. Q13Determine the vector product of $\vec{v_1} = 2\hat{i}+3\hat{j}-\hat{k}$ and $\vec{v_2} = \hat{i}+2\hat{j}-3\hat{k}$.Free
  4. Q14Given $\vec{v_1} = 5\hat{i}+2\hat{j}$ and $\vec{v_2} = a\hat{i}-6\hat{j}$ are perpendicular to each other, determine the value of $a$.Preview
  5. Q15Obtain derivatives of the following functions: (i) $x\sin x$ (ii) $x^4+\cos x$ (iii) $\dfrac{x}{\sin x}$Preview
  6. Q16Using the rule for differentiation for quotient of two functions, prove that $\dfrac{d}{dx}\left(\dfrac{\sin x}{\cos x}\right) = \sec^2 x$Preview
  7. Q17Evaluate the following integral: (i) $\displaystyle\int_0^{\pi/2} \sin x\, dx$ (ii) $\displaystyle\int_0^1 x\, dx$Preview