Notes-Part-1-Class-11-Science-Physics-Chapter-2-Mathematical Methods-Maharashtra Board

Mathematical Methods

Maharashtra Board-Class-11-Science-Physics-Chapter-2

Notes-Part-1

Topics to be Learn : Part-1

  • Vector analysis
  • Vector operations

Topics to be Learn : Part-2

  • Resolution of vectors
  • Multiplication of vectors

Introduction: Certain mathematical tools needs to understand the topics like Vector analysis and elementary calculus. In this chapter you are going to learn about vector analysis and a preliminary introduction to calculus which should be sufficient for you to understand the physics.

Vector Analysis:

Scalar quantity : A physical quantity that has only magnitude is called a scalar quantity or, simply, a scalar.

  • A scalar is thus completely specified by its magnitude — a number and unit, if any.
  • Examples : Mass, length, time, temperature, speed, energy, pressure, electric current, refractive index, Work.

Vector quantity : Physical quantities which need magnitude as well as direction for their complete description are called a vector quantity or, simply, a vector.

  • Examples : Force, displacement, velocity, acceleration, electric field intensity, magnetic induction, Momentum.

Graphical representation of Vector : A vector is represented graphically by an arrow pointing in the direction of the vector; the length of the arrow, for a given unit, represents the vector’s magnitude.

Symbolic representation of Vector : Symbolically, in formulae, a vector is represented by a letter with an arrow above it, e.g.\(\vec{X}\). The magnitude of \(\vec{X}\) is denoted by X or \(|\vec{X}|\). It may also be represented by the end-points of the vector,

e.g. \(\vec{OP}\) as in Fig. Here, O is called the tail of the vector and P is called the tip or head of the vector.

Examples of different types of vectors :

Examples of different types of vectors :

(i) Null or zero vector : A null or zero vector is defined to be a vector of zero magnitude.

So that for any vector \(\vec{X}\), \(\vec{X}\) + \(\vec{0}\)  = \(\vec{X}\); \(\vec{0}\) is called the additive identity.

  • A null vector is not a proper vector because we cannot speak of its direction.
  • Examples ; If a particle moves from A to B and then back to A (not necessarily along the same path), its net displacement is a null or zero vector.
  • Other examples of a zero vector are (i) the velocity of a stationary particle (ii) the acceleration of a particle moving with uniform velocity.

(ii) Resultant vector: The resultant of two or more vectors is that single vector, which produces the same effect, as produced by all the vectors together is called resultant vector.

  • The process of finding the resultant of two or more vectors is called the composition or addition of vectors.

(iii) Negative vector (opposite vector): A negative vector of a given vector is a vector of the same magnitude but opposite in direction to that of the given vector.

Figure shows a vector \(\vec{B}\) which is equal in magnitude to the vector \(\vec{A}\)  but oppositely directed.

Symbolically, we write \(\vec{B}\) = \(−\vec{A}\) . From the procedure for addition of two vectors and the definition of the null vector, it follows that  \(\vec{A} + (−\vec{A})\) = \(\vec{0}\), \(−\vec{A}\) is called the additive inverse of \(\vec{A}\).

We may also write \(\vec{B}\) = (— 1)\(\vec{A}\) , i.e., \(\vec{B}\) is a scalar multiple of \(\vec{A}\) in the special case when the scalar is the negative integer (— 1). By the same token, We may say \(\vec{A}\) = (—1)\(\vec{B}\) , i.e., \(\vec{A}\) is the negative of the vector \(\vec{B}\).

Never use a minus sign in a vector diagram unless it specifically designates the negative of a vector. That is, never add a minus sign to a vector just because it happens to point along the negative direction of a coordinate axis. For example, labelling the downward vector for the acceleration due to gravity \(−\vec{g}\)   is incorrect; it should just be labelled \(\vec{g}\) 

(iv) Equal vectors : Two vectors are said to be equal if they represent the same physical quantity and have the same magnitude and direction. 

  • Two vectors of the same type are parallel if they have the same direction; they are equal only when they also have the same magnitude.

(v) Position vector : A vector which gives the position of a particle at a point with respect to the origin of a chosen coordinate system is called the position vector of the particle.

In Fig   \(\vec{r}\) = \(\vec{OP}\)  is the position vector of the particle present at P.

(vi) Unit vector : A unit vector is a dimensionless vector with the magnitude 1 (pure number).

  • A unit vector is used only to specify a given direction or the direction of a vector quantity.
  • A unit vector is usually denoted by a lower case letter with a caret symbol ( ^ ) above it, e.g. \(\hat{u}\)  . For example, given a nonzero vector \(\vec{F}\), a unit vector \(\hat{u}\)   in its direction is given by the ratio of the vector to its magnitude \(\hat{u}=\frac{\vec{F}}{F}\)
  • Thus, the vector  \(\vec{F}\) = \(F\hat{u}\) 
  • Since a unit vector in the direction of a given vector is given by the ratio of the vector to its magnitude, the dimensions and units of a unit vector is none,

Rectangular unit vectors : The unit vectors specifying the positive directions of the Cartesian coordinate axes, x, y and z, are called the rectangular unit Vectors, they are usually labelled as \(\hat{i}\), \(\hat{j}\)  and \(\hat{k}\)  respectively, as shown in Fig.

Since a vector does not change when translated parallel to itself, it is not necessary that these unit vectors be located at the origin of the coordinate system or along the respective axis. They may be moved around preserving their directions.

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Q. Can the resultant of two vectors of unequal magnitudes be zero? Explain.

Answer :

No. If the resultant of two vectors is a zero vector, then one vector is the additive inverse of the other, i.e., they must be equal in magnitude and opposite in direction. If \(\vec{A}\) + \(\vec{B}\)  = 0, then \(\vec{B}\) = \(−\vec{A}\). ( \(\vec{B}\) is negative of \(\vec{A}\) ) or \(\vec{A}\) = \(−\vec{B}\) ( \(\vec{A}\) is negative of \(\vec{B}\) ).

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Vector Operations:

Multiplication of a Vector by a Scalar :

Multiplying a vector  by a scalar quantity, say n, yields another vector. \(\vec{Q}\) = \(n\vec{P}\) 

\(\vec{Q}\) will be a vector whose direction is the same as that of \(\vec{P}\) and magnitude is n times the magnitude of .

  • If n is positive, n\(\vec{P}\)  has the same direction as that of  \(\vec{Q}\)
  • while if n is negative, n\(\vec{P}\) has a direction opposite to that of \(\vec{Q}\) .

Examples of a vector, where a vector of different type is obtained when a vector is multiplied by a scalar :

Examples of a vector, where a vector of different type is obtained when a vector is multiplied by a scalar :

  • The product of time t (a scalar) and velocity \(\vec{v}\)  (a vector) gives displacement \(\vec{s}\)  = t \(\vec{Q}\) , which is a different physical quantity.
  •  \(\vec{P}\)  = m\(\vec{v}\), where \(\vec{P}\) is momentum of a particle, m is mass of the particle and \(\vec{v}\) is velocity of the particle.
  • \(\vec{F}\) = m\(\vec{a}\) , where \(\vec{F}\) is net force on a particle, m is mass of the particle and \(\vec{a}\) is acceleration of the particle.

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Q. Given two parallel vectors, show that one vector can be expressed as a scalar multiple of the other.

Answer :

Let \(\vec{A}\) and \(\vec{B}\) be two given parallel vectors, each parallel to a unit vector \(\hat{u}\),

where \(\hat{u}\) = \(\frac{\vec{A}}{A}= \frac{\vec{B}}{B}\)

∴  \(\vec{B}\) = B\(\hat{u}\) = B\(\frac{\vec{A}}{A} = λ\vec{A}\)

where the scalar multiple λ = \(\frac{B}{A}\).

Similarly,

\(\vec{A}\) = A\(\hat{u}\) = A\(\frac{\vec{B}}{B}\) = μ\(\vec{B}\)

where the scalar multiple μ = \(\frac{A}{B}\). 

Thus, given two parallel vectors, one vector can be expressed as a scalar multiple of the other.

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Addition and Subtraction of Vectors: The addition or subtraction of two or more vectors of the same type, i.e., describing the same physical quantity, gives rise to a single vector, such that the effect of this single vector is the same as the net effect of the vectors which have been added or subtracted.

Example :

Example: (i) Force \(\vec{F_1}\)  and \(\vec{F_2}\) force  can be added to give the resultant force

\(\vec{F}\) = \(\vec{F_1}\) + \(\vec{F_2}\)

But a force vector cannot be added to a velocity vector.

(ii) If vectors \(\vec{AB}\) and \(\vec{BC}\)  having the same or opposite direction but different magnitudes. The magnitude of their resultant is the addition of individual magnitudes, i.e., |\(\vec{AC}\)| = |\(\vec{AB}\)|+|\(\vec{BC}\)|  and direction of the resultant is the same as that of the individual vectors as shown in Fig

(iii) If the individual vectors are anti-parallel (i.e., in the opposite direction), the magnitude of their resultant is the difference of the individual magnitudes, and the direction is that of the larger vector i.e., |\(\vec{AC}\)| = |\(\vec{AB}\)|−|\(\vec{BC}\)|  as shown in Fig.

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Triangle law of vector addition : If two given vectors, representing the same physical quantity, are drawn in sequence, regardless of order but preserving their magnitudes and directions, to form two sides of a triangle, then the vector drawn from the tail of the first vector to the head of the second vector completing the triangle represents the resultant of the two given vectors.

Explanation :

Explanation : A vector is not changed by translation parallel to itself. Preserving the directions of two given vectors, (\vec{P}\) and (\vec{Q}\), the tail of one vector is joined to the head of the other, regardless of order, as shown in Fig. Then, the arrow from the tail of the first vector, to the head of the second gives the resultant \(\vec{R}\) = \(\vec{P}\) + \(\vec{Q}\)  in magnitude and direction

Since the order in which the vectors are drawn to represent the two sides of the triangle is immaterial, it at once follows that \(\vec{P}\) + \(\vec{Q}\) = \(\vec{Q}\) + \(\vec{P}\) , i.e., vector addition is commutative.

  • Dimensional homogeneity of an equation requires that two or more physical quantities to be added — all vectors or all scalars — must have the same dimensions and must also be of the same type.

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Q. Using the triangle law of vector addition, show that vector addition is associative :

Answer :

Given any three vectors, \(\vec{A}\), \(\vec{B}\) ,  and \(\vec{C}\), we may draw two sides OP and PQ of ΔOPQ in sequence such that \(\vec{OP}\) = \(\vec{A}\) and \(\vec{PQ}\) = \(\vec{B}\), as shown in Fig.

Fig-Associative law of vector addition

Then, by the triangle law of vector addition, \(\vec{OQ}\) = \(\vec{A}\)+\(\vec{B}\). Now, drawing the side QR of ΔOQR, such that \(\vec{QR}\) = \(\vec{C}\).

\(\vec{OR}\) = \(\vec{OQ}\)+\(\vec{QR}\) = (\(\vec{A}\)+\(\vec{B}\) ) + \(\vec{C}\)    …....(1)

\(\vec{PR}\) = \(\vec{PQ}\)+\(\vec{QR}\) = ( (\(\vec{B}\)+\(\vec{C}\) ) .....from  ΔPQR ….. (2)

\(\vec{OR}\) = \(\vec{OP}\)+\(\vec{PR}\) = \(\vec{A}\)+( \(\vec{B}\) + \(\vec{C}\) )  ....from  ΔOPR ….. (3)

∴ from  (1) and (3) \(\vec{OR}\) = (\(\vec{A}\)+\(\vec{B}\) ) + \(\vec{C}\)  = \(\vec{A}\)+( \(\vec{B}\) + \(\vec{C}\) )

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Q. Graphically show how a vector is subtracted from another vector using the triangle. Hence, explain the statement, ’Vector subtraction is not commutative.

Answer :

Subtraction of vectors : Given two vectors  \(\vec{A}\) and \(\vec{B}\) , to find \(\vec{C}\) = \(\vec{A}\) − \(\vec{B}\), we write

\(\vec{C}\) = \(\vec{A}\) − \(\vec{B}\) = \(\vec{A}\) + \(−\vec{B}\)

That is, from \(\vec{B}\) we draw its negative, −\(\vec{B}\) , This is then added to \(\vec{A}\) according to the triangle law of vector addition. Thus, as shown in Fig.

\(\vec{C}\) = \(\vec{A}\) − \(\vec{B}\).  Thus, subtracting a vector is the same as adding its additive inverse.

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Noncommutativity of vector subtraction : To find  \(\vec{D}\) = \(\vec{B}\) − \(\vec{A}\) , we add the negative of \(\vec{A}\)  to \(\vec{B}\), as shown in above Fig. Clearly,  \(\vec{C}\) and \(\vec{D}\) are different.

Writing  \(\vec{B}\) − \(\vec{A}\) = −( \(\vec{A}\) − \(\vec{B}\) ), we see that  \(\vec{D}\) = −\(\vec{C}\) i.e. \(\vec{D}\) is the negative of \(\vec{C}\). Hence, vector subtraction is not commutative.

∴ Subtraction of vectors is neither commutative nor associative.

Polygon law of vector addition :

Explanation :

Q. Explain how triangle law of vector addition can be extended to graphically add more than two vectors.

Ans. Given any three vectors \(\vec{A}\), \(\vec{B}\) and \(\vec{C}\) representing the same physical quantity, we may draw two sides OP and OQ of ΔOPQ in sequence such that  \(\vec{OP}\) = \(\vec{A}\) and \(\vec{PQ}\) = \(\vec{B}\)  = , as shown in Fig.

Then, by the triangle law of addition,  \(\vec{OQ}\) = \(\vec{A\) + \(\vec{B}\)

Now, drawing the side QR of ΔOQR, such that \(\vec{QR}\) = \(\vec{C\)

\(\vec{OR}\) = \(\vec{OQ\) + \(\vec{QR}\) = (\(\vec{A\) + \(\vec{B}\) ) + \(\vec{C\)

∴ \(\vec{D\) = \(\vec{OR}\) = \(\vec{A\) + \(\vec{B}\) + \(\vec{C\)  is the resultant of \(\vec{A}\), \(\vec{B}\) and \(\vec{C}\)

Therefore, if several given vectors representing the same physical quantity, are drawn in sequence, regardless of order but preserving their magnitudes and directions, to form the sides of a polygon, then the vector drawn from the tail of the first vector to the head of the last vector completing the polygon represents the resultant of the given vectors.

This is called the polygon law of vector addition.

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Parallelogram law of vector addition : If two vectors are joined at their tail ends to form two adjacent sides of a parallelogram, the diagonal drawn from the tail ends represents, in magnitude and direction, the resultant of the two vectors.

Explanation :

In Fig.  vector \(\vec{OA}\) = \(\vec{P}\) and vector (\vec{OB}\) = \(\vec{Q}\), represent two vectors originating from point O, inclined to each other at an angle θ.

The parallelogram OACB completed, then according to this law, the diagonal \vec{OC}\) = \(\vec{R}\)  represents the resultant vector.

To find the magnitude of (\vec{R}\) drop a perpendicular from C to reach OA (extended) at D. 

In right angled triangle ODC, by application by Pythagoras theorem,

OC2 = OD2+DC2 = (OA+AD)2 + DC2

OC2 = OA2+2OA.AD+AD2+DC2

In the right angled triangle ADC, by application of Pythagoras theorem

AD2+DC2=AC2

∴ OC2=OA2+2OA. AD+ AC2 --- (1)

Also,

\(\vec{OA}\) = \(\vec{P}\), \(\vec{AC}\) = \(\vec{OB}\) = \(\vec{Q}\)  and \(\vec{OC}\) = \(\vec{R}\)

In Δ ADC, cos θ = AD/AC

∴ AD = AC cos θ = Q cos θ

Substituting in Eq. (1)

R2 = P2 + Q2 + 2 P Q cos θ.

R =  \(\sqrt{P^2 + Q^2 + 2PQ\,cos\,θ}\)    …….(2)

Equation (2) gives us the magnitude of resultant vector

To find the direction of the resultant vector , we will have to find the angle (α) made by  with

In ΔODC, tan α = DC/OD = \(\frac{DC}{OA+OD}\)  …..(3)

From the figure, sin θ = DC/AC

DC = AC sin θ  = Q sin θ

Also,

AD = AC cos θ = Q cos θ

and OA = \(\vec{P}\)

Substituting in Eq. (3), we get

tan α = \(\frac{Q\,sin\,θ}{P+Q\,cos\,θ}\)

∴ α = \(tan^{-1}(\frac{Q\,sin\,θ}{P+Q\,cos\,θ})\)   …..(4)

Equation (4) gives us the direction of resultant vector

If β is the angle between \(\vec{R}\) and \(\vec{Q}\) , it can be. similarly derived that

β = \(tan^{-1}(\frac{P\,sin\,θ}{Q+P\,cos\,θ})\)

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Q. If | \(\vec{A}+\vec{B}\) | = | \(\vec{A}-\vec{B}\) |,  what can be said about the angle between \(\vec{A}\) and \(\vec{B}\)  ?

Answer :

Ans. Let θ be the angle between  and  so that the smaller angle between  and   is 180° − θ.

Then, | \(\vec{A}+\vec{B}\) |2 = A2+B2+2AB cos θ

And | \(\vec{A}-\vec{B}\) |2 = A2 + B2 − 2AB cos (180° − θ) = A2 + B2 − 2AB cos θ

Given that | \(\vec{A}+\vec{B}\) | = | \(\vec{A}-\vec{B}\) |, equating the right hand sides,

2AB cos θ = −2AB cos θ

∴ 2 cos θ = 0

where it is assumed that A ≠ 0 and B ≠ 0.

∴ cos θ = 0 or θ = 90°

i.e.  \(\vec{A}\) and \(\vec{B}\) are perpendicular to each other-

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