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      Class 12 MATHS – JEE

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      • Class 12
      • Class 12 MATHS – JEE
      CoursesClass 12MathsClass 12 MATHS – JEE
      • 1.Relations and Functions
        11
        • Lecture1.1
          Revision – Functions and its Types 38 min
        • Lecture1.2
          Revision – Functions Types 17 min
        • Lecture1.3
          Revision – Sum Related to Relations 04 min
        • Lecture1.4
          Revision – Sums Related to Relations, Domain and Range 22 min
        • Lecture1.5
          Cartesian Product of Sets, Relation, Domain, Range, Inverse of Relation, Types of Relations 34 min
        • Lecture1.6
          Functions, Intervals 39 min
        • Lecture1.7
          Domain 01 min
        • Lecture1.8
          Problem Based on finding Domain and Range 39 min
        • Lecture1.9
          Types of Real function 31 min
        • Lecture1.10
          Odd & Even Function, Composition of Function 32 min
        • Lecture1.11
          Chapter Notes – Relations and Functions
      • 2.Inverse Trigonometric Functions
        16
        • Lecture2.1
          Revision – Introduction, Some Identities and Some Sums 16 min
        • Lecture2.2
          Revision – Some Sums Related to Trigonometry Identities, trigonometry Functions Table and Its Quadrants 35 min
        • Lecture2.3
          Revision – Trigonometrical Identities-Some important relations and Its related Sums 16 min
        • Lecture2.4
          Revision – Sums Related to Trigonometrical Identities 18 min
        • Lecture2.5
          Revision – Some Trigonometric Identities and its related Sums 42 min
        • Lecture2.6
          Revision – Trigonometry Equations 44 min
        • Lecture2.7
          Revision – Sum Based on Trigonometry Equations 08 min
        • Lecture2.8
          Introduction to Inverse Trigonometry Function, Range, Domain, Question based on Principal Value 37 min
        • Lecture2.9
          Property -1 of Inverse trigo function 28 min
        • Lecture2.10
          Property -2 to 4 of Inverse trigo function 48 min
        • Lecture2.11
          Questions based on properties of Inverse trigo function 19 min
        • Lecture2.12
          Question based on useful substitution 27 min
        • Lecture2.13
          Numerical problems 19 min
        • Lecture2.14
          Numerical problems 24 min
        • Lecture2.15
          Numerical problems , introduction to Differentiation 44 min
        • Lecture2.16
          Chapter Notes – Inverse Trigonometric Functions
      • 3.Matrices
        9
        • Lecture3.1
          What is matrix 26 min
        • Lecture3.2
          Types of matrix 28 min
        • Lecture3.3
          Operations of matrices 28 min
        • Lecture3.4
          Multiplication of matrices 29 min
        • Lecture3.5
          Properties of a matrices 44 min
        • Lecture3.6
          Numerical problems 19 min
        • Lecture3.7
          Solution of simultaneous linear equation 28 min
        • Lecture3.8
          Solution of simultaneous / Homogenous linear equation 24 min
        • Lecture3.9
          Chapter Notes – Matrices
      • 4.Determinants
        6
        • Lecture4.1
          Introduction of Determinants 23 min
        • Lecture4.2
          Properties of Determinants 29 min
        • Lecture4.3
          Numerical problems 15 min
        • Lecture4.4
          Numerical problems 16 min
        • Lecture4.5
          Applications of Determinants 17 min
        • Lecture4.6
          Chapter Notes – Determinants
      • 5.Continuity
        7
        • Lecture5.1
          Introduction to continuity 28 min
        • Lecture5.2
          Numerical problems 17 min
        • Lecture5.3
          Numerical problems 22 min
        • Lecture5.4
          Basics of continuity 26 min
        • Lecture5.5
          Numerical problems 17 min
        • Lecture5.6
          Numerical problems 11 min
        • Lecture5.7
          Chapter Notes – Continuity and Differentiability
      • 6.Differentiation
        14
        • Lecture6.1
          Introduction to Differentiation 27 min
        • Lecture6.2
          Important formula’s 29 min
        • Lecture6.3
          Numerical problems 29 min
        • Lecture6.4
          Numerical problems 31 min
        • Lecture6.5
          Differentiation by using trigonometric substitution 21 min
        • Lecture6.6
          Differentiation of implicit function 21 min
        • Lecture6.7
          Differentiation of logarthmetic function 31 min
        • Lecture6.8
          Differentiation of log function 25 min
        • Lecture6.9
          Infinite series & parametric function 26 min
        • Lecture6.10
          Infinite series & parametric function 27 min
        • Lecture6.11
          Higher order derivatives 27 min
        • Lecture6.12
          Differentiation of function of a function 16 min
        • Lecture6.13
          Numerical problems 27 min
        • Lecture6.14
          Numerical problems 04 min
      • 7.Mean Value Theorem
        4
        • Lecture7.1
          Lagrange theorem 24 min
        • Lecture7.2
          Rolle’s theorem 20 min
        • Lecture7.3
          Lagrange theorem 24 min
        • Lecture7.4
          Rolle’s theorem 20 min
      • 8.Applications of Derivatives
        6
        • Lecture8.1
          Rate of change of Quantities 30 min
        • Lecture8.2
          Rate of change of Quantities 18 min
        • Lecture8.3
          Rate of change of Quantities 18 min
        • Lecture8.4
          Approximation 10 min
        • Lecture8.5
          Approximation 05 min
        • Lecture8.6
          Chapter Notes – Applications of Derivatives
      • 9.Increasing and Decreasing Function
        5
        • Lecture9.1
          Introduction 36 min
        • Lecture9.2
          Numerical Problem 26 min
        • Lecture9.3
          Numerical Problem 22 min
        • Lecture9.4
          Numerical Problem 22 min
        • Lecture9.5
          Numerical Problem 21 min
      • 10.Tangents and Normal
        3
        • Lecture10.1
          Introduction 34 min
        • Lecture10.2
          Numerical Problems 32 min
        • Lecture10.3
          Angle of intersection of two curves 27 min
      • 11.Maxima and Minima
        10
        • Lecture11.1
          Introduction 28 min
        • Lecture11.2
          Local maxima & Local Minima 27 min
        • Lecture11.3
          Numerical Problems 37 min
        • Lecture11.4
          Maximum & minimum value in closed interval 19 min
        • Lecture11.5
          Application of Maxima & Minima 10 min
        • Lecture11.6
          Application of Maxima & Minima 14 min
        • Lecture11.7
          Numerical Problems 17 min
        • Lecture11.8
          Numerical Problems 18 min
        • Lecture11.9
          Numerical Problems 15 min
        • Lecture11.10
          Numerical Problems 14 min
      • 12.Integrations
        19
        • Lecture12.1
          Introduction to Indefinite Integration 37 min
        • Lecture12.2
          Integration by substitution 25 min
        • Lecture12.3
          Numerical problems on Substitution 39 min
        • Lecture12.4
          Numerical problems on Substitution 04 min
        • Lecture12.5
          Integration various types of particular function (Identities) 31 min
        • Lecture12.6
          Integration by parts-1 18 min
        • Lecture12.7
          Integration by parts-2 10 min
        • Lecture12.8
          Integration by parts-2 16 min
        • Lecture12.9
          Integration by parts-2 08 min
        • Lecture12.10
          ILATE Rule 12 min
        • Lecture12.11
          Integration of some special function 07 min
        • Lecture12.12
          Integration of some special function 06 min
        • Lecture12.13
          Integration by substitution using trigonometric 14 min
        • Lecture12.14
          Evaluation of some specific Integration 12 min
        • Lecture12.15
          Evaluation of some specific Integration 29 min
        • Lecture12.16
          Integration by partial fraction 27 min
        • Lecture12.17
          Integration of some special function 11 min
        • Lecture12.18
          Numerical Problems based on partial fraction 20 min
        • Lecture12.19
          Chapter Notes – Integrals
      • 13.Definite Integrals
        11
        • Lecture13.1
          Introduction 24 min
        • Lecture13.2
          Properties of Definite Integration 19 min
        • Lecture13.3
          Numerical problem based on properties 22 min
        • Lecture13.4
          Area under the curve 16 min
        • Lecture13.5
          Area under the curve (Ellipse) 20 min
        • Lecture13.6
          Area under the curve (Parabola) 10 min
        • Lecture13.7
          Area under the curve (Parabola & Circle) 40 min
        • Lecture13.8
          Area bounded by lines 10 min
        • Lecture13.9
          Numerical problems 25 min
        • Lecture13.10
          Area under the curve (Circle ) 02 min
        • Lecture13.11
          Chapter Notes – Application of Integrals
      • 14.Differential Equations
        6
        • Lecture14.1
          Introduction to chapter 38 min
        • Lecture14.2
          Solution of D.E. – Variable separation methods 14 min
        • Lecture14.3
          Solution of D.E. – Variable separation methods 27 min
        • Lecture14.4
          Solution of D.E. – Second order 21 min
        • Lecture14.5
          Homogeneous D.E. 31 min
        • Lecture14.6
          Chapter Notes – Differential Equations
      • 15.Vectors
        12
        • Lecture15.1
          Introduction , Basic concepts , types of vector 34 min
        • Lecture15.2
          Position vector, distance between two points, section formula 44 min
        • Lecture15.3
          Numerical problem 02 min
        • Lecture15.4
          collinearity of points and coplanarity of vector 34 min
        • Lecture15.5
          Direction cosine 18 min
        • Lecture15.6
          Projection , Dot product, Cauchy- Schwarz inequality 24 min
        • Lecture15.7
          Numerical problem (dot product) 20 min
        • Lecture15.8
          Vector (Cross) product , Lagrange’s Identity 15 min
        • Lecture15.9
          Numerical problem (cross product) 38 min
        • Lecture15.10
          Numerical problem (cross product) 06 min
        • Lecture15.11
          Numerical problem (cross product) 22 min
        • Lecture15.12
          Chapter Notes – Vectors
      • 16.Three Dimensional Geometry
        7
        • Lecture16.1
          Introduction to 3D, axis in 3D, plane in 3D, Distance between two points 32 min
        • Lecture16.2
          Numerical problems , section formula , centroid of a triangle 35 min
        • Lecture16.3
          projection , angle between two lines 40 min
        • Lecture16.4
          Numerical Problem based on Direction ratio & cosine 02 min
        • Lecture16.5
          locus of any point 15 min
        • Lecture16.6
          Numerical Problem based on locus 16 min
        • Lecture16.7
          Chapter Notes – Three Dimensional Geometry
      • 17.Direction Cosine
        2
        • Lecture17.1
          Introduction 34 min
        • Lecture17.2
          Angle Between two vectors 25 min
      • 18.Plane
        3
        • Lecture18.1
          Introduction to plane , general equation of a plane , normal form 31 min
        • Lecture18.2
          Angle between two planes 30 min
        • Lecture18.3
          Distance of a point from a plane 29 min
      • 19.Straight Lines
        22
        • Lecture19.1
          Revision – Introduction, Equation of Line, Slope or Gradient of a line 24 min
        • Lecture19.2
          Revision – Sums Related to Finding the Slope, Angle Between two Lines 22 min
        • Lecture19.3
          Revision – Cases for Angle B/w two Lines, Different forms of Line Equation 23 min
        • Lecture19.4
          Revision – Sums Related Finding the Equation of Line 27 min
        • Lecture19.5
          Revision – Sums based on Previous Concepts of Straight line 32 min
        • Lecture19.6
          Revision – Parametric Form of a Straight Line 16 min
        • Lecture19.7
          Revision – Sums Related to Parametric Form of a Straight Line 17 min
        • Lecture19.8
          Revision – Sums Based on Concurrent of lines, Angle b/w Two Lines 45 min
        • Lecture19.9
          Revision – Different condition for Angle b/w two lines 04 min
        • Lecture19.10
          Revision – Sums Based on Angle b/w Two Lines 36 min
        • Lecture19.11
          Revision – Equation of Straight line Passes Through a Point and Make an Angle with Another Line 09 min
        • Lecture19.12
          Revision – Sums Based on Equation of Straight line Passes Through a Point and Make an Angle with Another Line 15 min
        • Lecture19.13
          Revision – Sums Based on Equation of Straight line Passes Through a Point and Make an Angle with Another Line 17 min
        • Lecture19.14
          Revision – Finding the Distance of a point from the line 35 min
        • Lecture19.15
          Revision – Sum Based on Finding the Distance of a point from the line and B/w Two Parallel Lines 33 min
        • Lecture19.16
          Introduction to straight line , symmetric form , Angle between the lines 27 min
        • Lecture19.17
          Numerical Problem 18 min
        • Lecture19.18
          Angle between two lines 32 min
        • Lecture19.19
          Unsymmetric form of Line 26 min
        • Lecture19.20
          Numerical problem , perpendicular distance of a point from a line 22 min
        • Lecture19.21
          Numerical Problem 21 min
        • Lecture19.22
          Numerical problem , Condition for a line lie on a plane 26 min
      • 20.Straight Lines (Vector)
        4
        • Lecture20.1
          Vector and Cartesian equation of a straight line 27 min
        • Lecture20.2
          Angle between two straight line 25 min
        • Lecture20.3
          Numerical problems 37 min
        • Lecture20.4
          Shortest Distance between two lines 22 min
      • 21.Linear Programming
        5
        • Lecture21.1
          Introduction to L.P. 30 min
        • Lecture21.2
          Numerical Problems 43 min
        • Lecture21.3
          Numerical Problems 23 min
        • Lecture21.4
          Numerical Problems 17 min
        • Lecture21.5
          Chapter Notes – Linear Programming
      • 22.Probability
        23
        • Lecture22.1
          Introduction to probability 41 min
        • Lecture22.2
          Types of events 42 min
        • Lecture22.3
          Numerical problems 30 min
        • Lecture22.4
          Conditional probability 12 min
        • Lecture22.5
          Numerical problems 09 min
        • Lecture22.6
          Numerical problems (conditional Probability) 04 min
        • Lecture22.7
          Numerical problems (conditional Probability) 06 min
        • Lecture22.8
          Numerical problems (conditional Probability) 05 min
        • Lecture22.9
          Numerical problems (conditional Probability) 06 min
        • Lecture22.10
          Bayes’ Theorem 17 min
        • Lecture22.11
          Numerical problem ( conditional Probability) 04 min
        • Lecture22.12
          Numerical problem ( Baye’s Theorem) 19 min
        • Lecture22.13
          Numerical problem ( Baye’s Theorem) 18 min
        • Lecture22.14
          Numerical problem ( Baye’s Theorem) 10 min
        • Lecture22.15
          Mean and Variance of a random variable 09 min
        • Lecture22.16
          Mean and Variance of a random variable 09 min
        • Lecture22.17
          Mean and Variance of a random variable 08 min
        • Lecture22.18
          Mean and Variance of a discrete random variable 07 min
        • Lecture22.19
          Numerical problem 18 min
        • Lecture22.20
          Bernoulli’s Trials & Binomial Distribution 11 min
        • Lecture22.21
          Numerical problem 12 min
        • Lecture22.22
          Mean and Variance of Binomial Distribution 05 min
        • Lecture22.23
          Chapter Notes – Probability
      • 23.Limits
        4
        • Lecture23.1
          Introduction to limits 35 min
        • Lecture23.2
          Numerical problems 27 min
        • Lecture23.3
          Rationalization 33 min
        • Lecture23.4
          Limits in trigonometry 29 min
      • 24.Partial Fractions
        4
        • Lecture24.1
          Introduction to partial fraction 27 min
        • Lecture24.2
          Partial Fractions 02 29 min
        • Lecture24.3
          Partial Fractions 03 17 min
        • Lecture24.4
          Improper partial fraction 20 min

        Chapter Notes – Vectors

        A vector has direction and magnitude both but scalar has only magnitude.
        The magnitude of a vector a is denoted by |a| or a. It is a non-negative scalar.

         

        Equality of Vectors

        Two vectors a and b are said to be equal written as a = b, if they have (i) same length (ii) the same or parallel support and (iii) the same sense.

        Types of Vectors

        (i) Zero or Null Vector

        A vector whose initial and terminal points are coincident is called zero or null vector. It is denoted by 0.

        (ii) Unit Vector

        A vector whose magnitude is unity is called a unit vector which is denoted by nˆ

        (iii) Free Vectors

        If the initial point of a vector is not specified, then it is said to be a free vector.

        (iv) Negative of a Vector

        A vector having the same magnitude as that of a given vector a and the direction opposite to that of a is called the negative of a and it is denoted by —a.

        (v) Like and Unlike Vectors

        Vectors are said to be like when they have the same direction and unlike when they have opposite direction.

        (vi) Collinear or Parallel Vectors

        Vectors having the same or parallel supports are called collinear vectors.

        (vii) Coinitial Vectors

        Vectors having same initial point are called coinitial vectors.

        (viii) Coterminous Vectors

        Vectors having the same terminal point are called coterminous vectors.

        (ix) Localized Vectors

        A vector which is drawn parallel to a given vector through a specified point in space is called localized vector.

        (x) Coplanar Vectors

        A system of vectors is said to be coplanar, if their supports are parallel to the same plane. Otherwise they are called non-coplanar vectors.

        (xi) Reciprocal of a Vector

        A vector having the same direction as that of a given vector but magnitude equal to the reciprocal of the given vector is known as the reciprocal of a. i.e., if |a| = a, then |a-1| = 1 / a.

         

        Addition of Vectors

        Let a and b be any two vectors. From the terminal point of a, vector b is drawn. Then, the vector from the initial point O of a to the terminal point B of b is called the sum of vectors a and b and is denoted by a + b. This is called the triangle law of addition of vectors.

        Addition of Vectors

        Parallelogram Law

        Let a and b be any two vectors. From the initial point of a, vector b is drawn and parallelogram OACB is completed with OA and OB as adjacent sides. The vector OC is efined as the sum of a and b. This is called the parallelogram law of addition of vectors.

        The sum of two vectors is also called their resultant and the process of addition as composition.

        Parallelogram Law

        Properties of Vector Addition

        (i) a + b = b + a (commutativity)
        (ii) a + (b + c)= (a + b)+ c (associativity)
        (iii) a+ O = a (additive identity)
        (iv) a + (— a) = 0 (additive inverse)
        (v) (k1 + k2) a = k1 a + k2a (multiplication by scalars)
        (vi) k(a + b) = k a + k b (multiplication by scalars)
        (vii) |a+ b| ≤ |a| + |b| and |a – b| ≥ |a| – |b|

        Difference (Subtraction) of Vectors

        If a and b be any two vectors, then their difference a – b is defined as a + (- b).

        Difference (Subtraction) of Vectors

        Multiplication of a Vector by a Scalar

        Let a be a given vector and λ be a scalar. Then, the product of the vector a by the scalar λ is λ a and is called the multiplication of vector by the scalar.

        Important Properties

        (i) |λ a| = |λ| |a|
        (ii) λ O = O
        (iii) m (-a) = – ma = – (m a)
        (iv) (-m) (-a) = m a
        (v) m (n a) = mn a = n(m a)
        (vi) (m + n)a = m a+ n a
        (vii) m (a+b) = m a + m b

        Vector Equation of Joining by Two Points

        Let P1 (x1, y1, z1) and P2 (x2, y2, z2) are any two points, then the vector joining P1 and P2 is the vector P1 P2.

        Vector Equation of Joining by Two Points

        The component vectors of P and Q are

        OP = x1i + y1j + z1k and OQ = x2i + y2j + z2k
        i.e., P1 P2 = (x2i + y2j + z2k) – (x1i + y1j + z1k)
        = (x2 – x1) i + (y2 – y1) j + (z2 – z1) k

        Its magnitude is P1 P2 = √(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2

        Position Vector of a Point

        The position vector of a point P with respect to a fixed point, say O, is the vector OP. The fixed point is called the origin.
        Let PQ be any vector. We have PQ = PO + OQ = — OP + OQ = OQ — OP = Position vector of Q — Position vector of P.

        Position Vector of a Point

        i.e., PQ = PV of Q — PV of P

        Collinear Vectors

        Vectors a and b are collinear, if a = λb, for some non-zero scalar λ.

         

        Collinear Points

        Let A, B, C be any three points.
        Points A, B, C are collinear <=> AB, BC are collinear vectors.
        <=> AB = λBC for some non-zero scalar λ.

        Section Formula

        Let A and B be two points with position vectors a and b, respectively and OP= r.
        (i) Let P be a point dividing AB internally in the ratio m : n. Then,
        r = m b + n a / m + n

        Section Formula

        Also, (m + n) OP = m OB + n OA

        (ii) The position vector of the mid-point of a and b is a + b / 2.
        (iii) Let P be a point dividing AB externally in the ratio m : n. Then, r = m b + n a / m + n

        Position Vector of Different Centre of a Triangle

        (i) If a, b, c be PV’s of the vertices A, B, C of a ΔABC respectively, then the PV of the centroid G of the triangle is a + b + c / 3.
        (ii) The PV of incentre of ΔABC is (BC)a + (CA)b + (AB)c / BC + CA + AB
        (iii) The PV of orthocentre of ΔABC is a(tan A) + b(tan B) + c(tan C) / tan A + tan B + tan C

        Scalar Product of Two Vectors

        If a and b are two non-zero vectors, then the scalar or dot product of a and b is denoted by a * b and is defined as a * b = |a| |b| cos θ, where θ is the angle between the two vectors and 0 < θ < π .

        (i) The angle between two vectors a and b is defined as the smaller angle θ between them, when they are drawn with the same initial point. Usually, we take 0 < θ < π.Angle between two like vectors is O and angle between two unlike vectors is π .
        (ii) If either a or b is the null vector, then scalar product of the vector is zero.
        (iii) If a and b are two unit vectors, then a * b = cos θ.
        (iv) The scalar product is commutative i.e., a * b= b * a
        (v) If i , j and k are mutually perpendicular unit vectors i , j and k, then i * i = j * j = k * k =1 and i * j = j * k = k * i = 0
        (vi) The scalar product of vectors is distributive over vector addition.
        (a) a * (b + c) = a * b + a * c (left distributive)
        (b) (b + c) * a = b * a + c * a (right distributive)

        Note Length of a vector as a scalar product If a be any vector, then the scalar product a * a = |a| |a| cosθ ⇒ |a|2 = a2 ⇒ a = |a| Condition of perpendicularity a * b = 0 <=> a ⊥ b, a and b being non-zero vectors.

        Important Points to be Remembered

        (i) (a + b) * (a – b) = |a|22 – |b|2
        (ii) |a + b|2 = |a|22 + |b|2 + 2 (a * b)
        (iii) |a – b|2 = |a|22 + |b|2 – 2 (a * b)
        (iv) |a + b|2 + |a – b|2 = (|a|22 + |b|2) and |a + b|2 – |a – b|2 = 4 (a * b) or a * b = 1 / 4 [ |a + b|2 – |a – b|2 ]
        (v) If |a + b| = |a| + |b|, then a is parallel to b.
        (vi) If |a + b| = |a| – |b|, then a is parallel to b.
        (vii) (a * b)2 ≤ |a|22 |b|2
        (viii) If a = a1i + a2j + a3k, then |a|2 = a * a = a12 + a2 2 + a3 2 Or |a| = √a1 2 + a2 2 + a3 2
        (ix) Angle between Two Vectors
        If θ is angle between two non-zero vectors, a, b, then we have a * b = |a| |b| cos θ cos θ = a * b / |a| |b| If a = a1i + a2j + a3k and b = b1i + b2j + b3k
        Then, the angle θ between a and b is given by cos θ = a * b / |a| |b| = a1b1 + a2b2 + a3b3 / √a1 2 + a2 2 + a3 2 √b1 2 + b2 2 + b3 2
        (x) Projection and Component of a Vector
        Projection of a on b = a * b / |a|
        Projection of b on a = a * b / |a|
        Vector component of a vector a on b

        Projection and Component of a Vector

        Similarly, the vector component of b on a = ((a * b) / |a2|) * a

        (xi) Work done by a Force

        The work done by a force is a scalar quantity equal to the product of the magnitude of the force and the resolved part of the displacement.

        ∴ F * S = dot products of force and displacement.

        Suppose F1, F1,…, Fn are n forces acted on a particle, then during the displacement S of the particle, the separate forces do quantities of work F1 * S, F2 * S, Fn * S.

        Work done by a Force

        Here, system of forces were replaced by its resultant R.

         

        Vector or Cross Product of Two Vectors

        The vector product of the vectors a and b is denoted by a * b and it is defined as

        a * b = (|a| |b| sin θ) n = ab sin θ n …..(i)

        where, a = |a|, b= |b|, θ is the angle between the vectors a and b and n is a unit vector which is perpendicular to both a and b, such that a, b and n form a right-handed triad of vectors.

        Important Points to be Remembered

        (i) Let a = a1i + a2j + a3k and b = b1i + b2j + b3k

        Important Points to be Remembered

        (ii) If a = b or if a is parallel to b, then sin θ = 0 and so a * b = 0.
        (iii) The direction of a * b is regarded positive, if the rotation from a to b appears to be anticlockwise.
        (iv) a * b is perpendicular to the plane, which contains both a and b. Thus, the unit vector perpendicular to both a and b or to the plane containing is given by n = a * b / |a * b| = a * b / ab sin θ
        (v) Vector product of two parallel or collinear vectors is zero.
        (vi) If a * b = 0, then a = O or b = 0 or a and b are parallel on collinear.
        (vii) Vector Product of Two Perpendicular Vectors If θ = 900, then sin θ = 1, i.e. , a * b = (ab)n or |a * b| = |ab n| = ab
        (viii) Vector Product of Two Unit Vectors
        If a and b are unit vectors, then
        a = |a| = 1, b = |b| = 1
        ∴ a * b = ab sin θ n = (sin theta;).n
        (ix) Vector Product is not Commutative
        The two vector products a * b and b * a are equal in magnitude but opposite in direction.
        i.e., b * a =- a * b ……..(i)
        (x) The vector product of a vector a with itself is null vector, i. e., a * a= 0.
        (xi) Distributive Law For any three vectors a, b, c a * (b + c) = (a * b) + (a * c)
        (xii) Area of a Triangle and Parallelogram
        (a) The vector area of a ΔABC is equal to 1 / 2 |AB * AC| or 1 / 2 |BC * BA| or 1 / 2 |CB * CA|.
        (b) The area of a ΔABC with vertices having PV’s a, b, c respectively, is 1 / 2 |a * b + b * c + c * a|.
        (c) The points whose PV’s are a, b, c are collinear, if and only if a * b + b * c + c * a
        (d) The area of a parallelogram with adjacent sides a and b is |a * b|.
        (e) The area of a Parallelogram with diagonals a and b is 1 / 2 |a * b|.
        (f) The area of a quadrilateral ABCD is equal to 1 / 2 |AC * BD|.
        (xiii) Vector Moment of a Force about a Point

        The vector moment of torque M of a force F about the point O is the vector whose magnitude is equal to the product of |F| and the perpendicular distance of the point O from the line of action of F.

        Vector Moment of a Force about a Point

        ∴ M = r * F

        where, r is the position vector of A referred to O.

        (a) The moment of force F about O is independent of the choice of point A on the line of action of F.
        (b) If several forces are acting through the same point A, then the vector sum of the moments of the separate forces about a point O is equal to the moment of their resultant force about O.
        (xiv) The Moment of a Force about a Line

        The Moment of a Force about a Line

        Let F be a force acting at a point A, O be any point on the given line L and a be the unit vector along the line, then moment of F about the line L is a scalar given by (OA x F) * a
        (xv) Moment of a Couple
        (a) Two equal and unlike parallel forces whose lines of action are different are said to constitute a couple.
        (b) Let P and Q be any two points on the lines of action of the forces – F and F, respectively.

        Moment of a Couple

        The moment of the couple = PQ x F

        Scalar Triple Product

        If a, b, c are three vectors, then (a * b) * c is called scalar triple product and is denoted by [a b c].
        ∴ [a b c] = (a * b) * c

         

        Geometrical Interpretation of Scalar Triple Product

        The scalar triple product (a * b) * c represents the volume of a parallelepiped whose coterminous edges are represented by a, b and c which form a right handed system of vectors.

        Expression of the scalar triple product (a * b) * c in terms of components
        a = a1i + a1j + a1k, b = a2i + a2j + a2k, c = a3i + a3j + a3k is

        Geometrical Interpretation of Scalar Triple Product

        Properties of Scalar Triple Products

        1. The scalar triple product is independent of the positions of dot and cross i.e., (a * b) * c = a * (b * c).

        2. The scalar triple product of three vectors is unaltered so long as the cyclic order of the vectors remains unchanged.
        i.e., (a * b) * c = (b * c) * a= (c * a) * b or [a b c] = [b c a] = [c a b]

        .3. The scalar triple product changes in sign but not in magnitude, when the cyclic order is changed.
        i.e., [a b c] = – [a c b] etc.

        4. The scalar triple product vanishes, if any two of its vectors are equal.
        i.e., [a a b] = 0, [a b a] = 0 and [b a a] = 0.

        5. The scalar triple product vanishes, if any two of its vectors are parallel or collinear.

        6. For any scalar x, [x a b c] = x [a b c]. Also, [x a yb zc] = xyz [a b c].

        7. For any vectors a, b, c, d, [a + b c d] = [a c d] + [b c d]

        8. [i j k] = 1

        Properties of Scalar Triple Products

        11. Three non-zero vectors a, b and c are coplanar, if and only if [a b c] = 0.

        12. Four points A, B, C, D with position vectors a, b, c, d respectively are coplanar, if and only if [AB AC AD] = 0.
        i.e., if and only if [b — a c— a d— a] = 0.

        13. Volume of parallelepiped with three coterminous edges a, b,c is | [a b c] |.

        14. Volume of prism on a triangular base with three coterminous edges a,b,c is 1/2 | [a b c] |.

        15. Volume of a tetrahedron with three coterminous edges a, b,c is 1 / 6 | [a b c] |.

        16. If a, b, c and d are position vectors of vertices of a tetrahedron, then Volume = 1 / 6 [b — a c — a d — a].

        Vector Triple Product

        If a, b, c be any three vectors, then (a * b) * c and a * (b * c) are known as vector triple product.
        ∴ a * (b * c)= (a * c)b — (a * b) c and (a * b) * c = (a * c)b — (b * c) a

        Important Properties

        (i) The vector r = a * (b * c) is perpendicular to a and lies in the plane b and c.

        (ii) a * (b * c) ≠ (a * b) * c, the cross product of vectors is not associative.

        (iii) a * (b * c)= (a * b) * c, if and only if and only if (a * c)b — (a * b) c = (a * c)b — (b * c) a, if and only if c = (b * c) / (a * b) * a Or if and only if vectors a and c are collinear.

        Reciprocal System of Ve ctors

        Let a, b and c be three non-coplanar vectors and let a’ = b * c / [a b c], b’ = c * a / [a b c], c’ = a * b / [a b c]
        Then, a’, b’ and c’ are said to form a reciprocal system of a, b and c.

        Properties of Reciprocal System

        (i) a * a’ = b * b’= c * c’ = 1
        (ii) a * b’= a * c’ = 0, b * a’ = b * c’ = 0, c * a’ = c * b’= 0
        (iii) [a’, b’, c’] [a b c] = 1 ⇒ [a’ b’ c’] = 1 / [a b c]
        (iv) a = b’ * c’ / [a’, b’, c’], b = c’ * a’ / [a’, b’, c’], c = a’ * b’ / [a’, b’, c’]
        Thus, a, b, c is reciprocal to the system a’, b’ ,c’.
        (v) The orthonormal vector triad i, j, k form self reciprocal system.
        (vi) If a, b, c be a system of non-coplanar vectors and a’, b’, c’ be the reciprocal system of vectors, then any vector r can be expressed as r = (r * a’ )a + (r * b’)b + (r * c’)c.

        Linear Combination of Vectors

        Let a, b, c,… be vectors and x, y, z, … be scalars, then the expression x a yb + z c + … is called a linear combination of vectors a, b, c,….

        Collinearity of Three Points

        The necessary and sufficient condition that three points with PV’s b, c are collinear is that there exist three scalars x, y, z not all zero such that xa + yb + zc ⇒ x + y + z = 0.

        Coplanarity of Four Points

        The necessary and sufficient condition that four points with PV’s a, b, c, d are coplanar, if there exist scalar x, y, z, t not all zero, such that xa + yb + zc + td = 0 rArr; x + y + z + t = 0.
        If r = xa + yb + zc…

        Then, the vector r is said to be a linear combination of vectors a, b, c,….

        Linearly Independent and Dependent System of Vectors

        (i) The system of vectors a, b, c,… is said to be linearly dependent, if there exists a scalars x, y, z, … not all zero, such that xa + yb + zc + … = 0.
        (ii) The system of vectors a, b, c, … is said to be linearly independent, if xa + yb + zc + td = 0 rArr; x + y + z + t… = 0.

        Important Points to be Remembered

        (i) Two non-collinear vectors a and b are linearly independent.
        (ii) Three non-coplanar vectors a, b and c are linearly independent.
        (iii) More than three vectors are always linearly dependent.

         

        Resolution of Components of a Vector in a Plane

        Let a and b be any two non-collinear vectors, then any vector r coplanar with a and b, can be uniquely expressed as r = x a + y b, where x, y are scalars and x a, y b are called components of vectors in the directions of a and b, respectively.

        Resolution of Components of a Vector in a Plane

        ∴ Position vector of P(x, y) = x i + y j.
        OP2 = OA2 + AP2 = |x|2 + |y|2 = x2 + y2
        OP = √x2 + y2. This is the magnitude of OP.

        where, x i and y j are also called resolved parts of OP in the directions of i and j, respectively.

        Vector Equation of Line and Plane

        (i) Vector equation of the straight line passing through origin and parallel to b is given by r = t b, where t is scalar.
        (ii) Vector equation of the straight line passing through a and parallel to b is given by r = a + t b, where t is scalar.
        (iii) Vector equation of the straight line passing through a and b is given by r = a + t(b – a), where t is scalar.
        (iv) Vector equation of the plane through origin and parallel to b and c is given by r = s b + t c, where s and t are scalars.
        (v) Vector equation of the plane passing through a and parallel to b and c is given by r = a + sb + t c, where s and t are scalars.
        (vi) Vector equation of the plane passing through a, b and c is r = (1 – s – t)a + sb + tc, where s and t are scalars.

        Bisectors of the Angle between Two Lines

        (i) The bisectors of the angle between the lines r = λa and r = μb are given by r = &lamba; (a / |a| &plumsn; b / |b|)
        (ii) The bisectors of the angle between the lines r = a + λb and r = a + μc are given by r = a + &lamba; (b / |b| &plumsn; c / |c|)

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