Vectors

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A presentation to explain the fundamentals of vectors.

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VECTORS: 

VECTORS

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Scalar and Vector Quantities Scalar quantities : Quantities which have only magnitude and no direction are called scalar quantities, e.g. mass, distance, time, speed, volume, density, pressure, work, energy, electric current, temperature, etc. Vector quantities : Quantities which have magnitude as well as direction and obey the triangle law of vector addition or equivalently the parallelogram law of vector addition are called vector quantities, e.g. position, displacement, velocity, force, acceleration, weight, momentum, impulse, electric field, magnetic field, current density, etc.

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Representation of vector quantities A vector quantity can be represented by an arrow. This arrow is called the ‘vector.’ The length of the arrow represents the magnitude and the tip of the arrow represents the direction. If a car A runs with a velocity of 10 m/s towards east; and another car B runs with a velocity of 20 m/s towards north-east. These velocities can be represented by vectors shown in the adjoining figure, taking each unit of length on the arrow to represent 5 m/s. N S E W Velocity of car A Velocity of car B

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Equality of Vectors Two or more vectors are said to be equal if, and only if, they have the same magnitude and same direction. In the adjoining figure, A, B and C are equal vectors. If the direction of a vector is reversed, the sign of the vector is reversed. This new vector is called the “negative vector” of the original vector. Here, the vector D is the negative vector of vectors A, B and C. Thus, A = B = C = -D A B C D

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Addition of Two Vectors Consider two vectors A and B. First vector A is drawn. Then starting from the arrow-head of A, the vector B is drawn. Now draw a vector R, starting from the initial point of A and ending at the arrow-head of B. Vector R would be the sum of A and B. R = A + B The magnitude of A + B can be determined by measuring the length of R and the direction can be expressed by measuring the angle between R and A (or B ) . A B R = A + B We can start drawing from vector B also, instead of vector A, as shown below:- A B R’ = B + A

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Addition of Two Vectors (contd.) The vectors R and R’ obtained in the previous slide are parallel to each other and their lengths and directions are same. Hence, R = R’ ∴ A + B = B + A Thus, addition of vectors is commutative. This method of vector addition is called the method of triangle of vectors.

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Addition of Two Vectors (contd.) There is another method of adding two vectors, known as the “method of parallelogram of vectors.” According to this method, sum of two vectors A and B is a vector R represented by the diagonal of a parallelogram whose adjacent sides are represented by vectors A and B. A B R = A + B

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Addition of Two Vectors (contd.) The magnitude of the sum of two vectors depends upon the angle between the vectors. In the adjoining figure, two vectors A and B are added by changing the angle between them, keeping their magnitudes unchanged. It is seen that the sum R of A and B is maximum when A and B are parallel, i.e., when the angle between them is 0. The magnitude of R would be (A+B). When the angle between A and B is 180º, then magnitude of resultant vector R is minimum, equal to (A-B) if A is greater, or (B-A) if B is greater. A R B A B R A B R A B R Since the minimum magnitude of A + B is (A-B), hence two vectors of “different” magnitudes cannot be added to get a zero resultant.

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Addition of more than Two Vectors If more than two vectors are to be added, then we first determine the sum of any two vectors. The third vector is then added to this sum and this method is continued. Suppose we have to add four vectors A, B, C and D as in the adjoining figure. Then we proceed as follows:- R = (A + B) + C + D R = (E + C) + D R = F + D A B C D A B C E F D R The sum of vectors in each case is the v ector drawn to complete the polygon formed by the given vectors. Hence this method of addition of vectors is called “polygon method.”

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** The vectors need not be added in the order seen in the last slide. Vector C may be first added to vector A, then vector D and finally vector B. ∴ R’ = A + C + D + B But vector R and vector R’ are parallel, equal in length and are in the same direction. ∴ R = R’ o r, A + B + C + D = A + C + D + B Hence vector addition is associative. ** If three or more vectors themselves complete a triangle or a polygon, then their sum-vector or resultant vector cannot be drawn. It means that the sum of these vectors is zero. Addition of more than Two Vectors

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Analytical Method of Vector Addition Triangle Law of Vector Addition:- This law states that if two vectors are represented in magnitude and direction by the two sides of a triangle taken in the same order, then their resultant is represented by the third side of the triangle taken in the opposite order. Let two vectors A and B be represented, both in magnitude and direction, by the sides OP and PQ of a triangle OPQ taken in the same order. Then the resultant R will be represented by the closing side OQ taken in the opposite order. O P Q R A B E θ Φ

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Analytical Method of Vector Addition To find the magnitude of resultant R, a perpendicular QE from Q on side OP produced is drawn. Let ∠QPE = θ . Then, in right-angled △OEQ, we have:- OQ² = OE² + QE² = (OP + PE)² + QE² = OP² + PE² + 2.OP.PE + QE² Now, PE² + QE² = PQ² ∴ OQ² = OP² + PQ² + 2.OP.PE In right-angled △PEQ, we have cos θ = ∴ PE = PQ.cos θ ∴ OQ² = OP² + PQ² + 2.OP.PQ.cos θ ∴ R² = A² + B² + 2ABcos θ PE PQ R = √(A² + B² + 2ABcos θ )

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Analytical Method of Vector Addition To find out the direction of the resultant, suppose the resultant R makes an angle Φ with the direction of vector A. Then, tan Φ = = Now OP = A and PE = Bcos θ . To find QE, we consider △PEQ. We have:- s in θ = , or, QE = PQ sin θ = B sin θ . ∴ tan Φ = QE O E QE OP + PE QE PQ B sin θ A + B cos θ

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Analytical Method of Vector Addition (ii) Parallelogra m Law of Vector Addition:- This law states that if two vectors are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram drawn from the same point. Let two vectors A and B inclined to each other at an angle θ be represented in magnitude and direction, by the sides OP and OS of a parallelogram OPQS. Then, according to parallelogram law, the resultant of A and B is represented both in magnitude and direction by the diagonal OQ of the parallelogram. B A R O P E Q S θ Φ θ

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Analytical Method of Vector Addition As discussed in case of triangle law of vector addition, the magnitude and direction of the resultant R will be given by :- tan Φ = R = √(A² + B² + 2ABcos θ ) B sin θ A + B cos θ

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Special Cases (i) When two vectors are in the same direction : Then, θ = 0 so that cos θ = cos 0º = 1 and sin θ = sin 0º = 0. Then we have :- R = √(A ² + B² + 2AB.cos 0º ) = (A+B), and tan Φ = = 0, i.e., Φ = 0. Thus, the resultant R has a magnitude equal to the sum of the magnitudes of the vectors A and B and acts along the direction of A and B. (ii) When two vectors are at right angle to each other : Then, θ = 90º so that cos 90º = 0 and sin 90º = 1. Then, R = √(A ² + B² + 2AB.cos 90º ) = √(A ² + B ² ) , and tan Φ = = B x 0 A + B B sin 90º A + B cos 90º B A

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Special Cases (contd.) (iii) When two vectors are in opposite directions : Then, θ = 180º, so that cos 180º = -1 and sin 180º = 0. ∴ R = √(A² + B² + 2AB.cos 180º) = √(A-B)² = (A-B) or (B-A), and tan Φ = = 0, i.e., Φ = 0º or 180º. Thus, the magnitude of the resultant vector is equal to the difference of the magnitudes of the two vectors and acts in the direction of the bigger vector. Note:- The magnitude of the resultant of two vectors is maximum when they are in the same direction, and minimum when they are in opposite directions. B sin 180º A + B cos 180º

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Special Cases (contd.) Further in a parallelogram, if one diagonal is the sum of two adjacent sides, then the other diagonal is equal to its differences. In the adjoining figure, OQ = OP + PQ PS = PQ + QS But, QS = –OP Thus, PS = PQ – OP

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Subtraction of Vectors Suppose A and B are two vectors and the vector B is to be subtracted from vector A. The subtraction of B from A is same as addition of –B to A, i.e., A – B = A + (–B). Hence, to subtract vector B from A, first we reverse B to get –B. Then the vector –B is added to vector A. For this, we first draw vector A and then starting from the arrow-head of A, we draw the vector –B, and finally we draw a vector R from the initial point of A to the arrow-head of –B. Thus, vector R is the sum of A and –B, i.e., the difference A – B :- R = A + (–B) = A – B. A B A -B A -B R = A – B

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Multiplication of a Vector by a Scalar On multiplying a vector A by a scalar or a number k , a vector R (say) is obtained :- R = k A The magnitude of R is k times the magnitude of A and the direction of R is same as that of A. If k is a pure number having no unit, then the unit of R will be same as that of A. If a vector A is 5 cm long and directed towards east, then vector 2 A would be 10 cm long and directed towards east; and the vector -2 A would be 10 cm long but directed towards west.

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Multiplication of a Vector by a Scalar (contd.) If k is a physical quantity having a unit, then the unit of R will be obtained by multiplying the units of k and A. In this case, the vector R will represent a new physical quantity. For example, if we multiply a vector v (velocity) by a scalar m (mass) then their multiplication p (say) will represent a new vector quantity called momentum :- p = m v The unit of mass m is kg and the unit of velocity v is m/s. Hence, the unit of p will be kg.m/s. The direction of p will be same as that of v.

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Position and Displacement Vectors To describe the motion of an object in a plane, we use the concept of position and displacement vectors. For this, we select a point in the plane as origin and describe the position of the object with respect to that origin. Y O X P Q r 1 r 2 Suppose, at an instant of time t 1 , the object is at a point P in the X-Y plane of a cartesian coordinate system. Then a vector OP drawn from origin O to the point P is called the position vector of the object at time t 1 . It may be written as r 1 , where r 1 is the distance of the point P from the origin O. If the object moves to a point Q at time t 2 , then OQ or r 2 is the position vector of the object at time t 2 , where r 2 is the distance of the point Q from the origin O.

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Position and Displacement Vectors (contd.) Y O X P Q r 1 r 2 The vector PQ drawn from the point P to the point Q is the displacement vector of the object during the interval t 2 – t 1 . The vector PQ is the vector difference OQ – OP (since OP + PQ = OQ by triangle law of vector addition), i.e., PQ = r 2 – r 1 Thus, the displacement vector is the difference between the final and the initial position vectors.

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Zero Vector or Null Vector If two vectors A and B are equal, then their difference A – B is defined as zero vector or null vector and is denoted as 0. A – B = 0, if A = B. Thus, zero vector is a vector of zero magnitude having no specific direction . Its initial and terminal points are coincident. Properties:- A + 0 = A n 0 = 0 0 A = 0

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Need for a Zero Vector Though a zero vector does not quite fit in our description of a vector as it has no specific direction, in this way it is considered as one of the non-proper vectors. Still it is needed in vector algebra due to the following reasons :- What is A – B when A = B? What is A + B + C if these vectors form a closed figure? We know that with respect to origin in cartesian coordinate system, position vector of a point P is OP, then what’s the position vector of origin itself? What is the displacement vector of a stationary object? What is the acceleration vector of an object moving with a constant velocity? Answer to all these questions is a zero vector .

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Unit Vector A vector whose magnitude is unity is called a “unit vector”. If A is a vector whose magnitude A ≠ 0, then A / A is a unit vector whose direction is the direction of A. The unit vector in the direction of A is written as A. Thus, A = or, A = A A Thus, any vector in the direction of unit vector may be written as the product of the unit vector and the scalar magnitude of that vector. ^ ^ A A ^ Orthogonal Unit Vectors:- The unit vectors along the X-axis, Y-axis and Z-axis of the right-handed cartesian coordinate system are written as i, j, and k respectively. These are called orthogonal unit vectors. ^ ^ ^ Y X O Z j ^ i ^ ^ k

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Resolution of a Vector The resolution of a vector is opposite to vector addition. If a vector is resolved into two vectors whose combined effect is the same as that of the given vector, then the resolved vectors are called the “components” of the given vector. If a vector is resolved into two vectors which are mutually perpendicular, then these vectors are called the “rectangular components” of the given vector. Let us suppose that a given vector A is to be resolved into two rectangular components. For this, taking the initial point of A as origin O, rectangular axes OX and OY are drawn. Then perpendiculars are dropped on OX and OY from the arrow-head of A. These perpendiculars intersect OX and OY at P and Q respectively. Then the vectors A x and A y drawn from O to P and Q are the rectangular components of vector A. From rectangle OQRP, it is clear that the vector A is the sum of vectors A x and A y :- A = A x + A y . By measuring OP and OQ, the magnitudes of A x and A y can be determined. X O Y P R Q θ A A x A y

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Resolution of a Vector (contd.) Now let i and j be unit vectors along X and Y axes respectively, and A x and A y the scalar magnitudes of A x and A y respectively. Then, we may write :- A x = A x i and A y = A y j Thus, we have :- A = A x i + A y j This is the equation for vector A in terms of its rectangular components in a plane. If the vector A makes an angle θ with the X-axis, then we have :- A x = A cos θ and A y = A sin θ From these equations, we have :- A = √(A x ² + A y ²) θ = tan - 1 (A y / A x ) Thus, if we know the magnitudes A x and A y of the rectangular components of A, then from above two equations, we can determine respectively the magnitude and direction of vector A. ^ ^ ^ ^ ^ ^

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Multiplication of two Vectors The multiplication of two vector quantities cannot be done by simple algebraic method. The product of two vectors may be a scalar as well as a vector. For example, both ‘force’ and ‘displacement’ are vector quantities. Their product may be ‘work’ as well as ‘moment of force’. Work is a scalar but moment of force is a vector quantity. Vector quantities are represented by vectors. If the product of two vectors is a scalar quantity, then it is called ‘scalar product’ ; if the product is a vector quantity then it is called ‘vector product.’ If A and B are two vectors, then their scalar product is written as A ∙ B (read A dot B), and the vector product is written as A x B (read A cross B). Hence, the scalar product is also called ‘dot product’ and the vector product is also called ‘cross product.’

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Scalar/Dot Product of two Vectors The scalar product of two vectors is defined as a scalar quantity equal to the product of their magnitudes and the cosine of the angle between them. Thus, if θ is the angle between A and B, then, A ∙ B = A B cos θ , where A and B are the magnitudes of A and B. The quantity AB cos θ is a scalar quantity. θ A B Now, B cos θ is the component of vector B in the direction of A. Hence, the scalar product of two vectors is equal to the product of the magnitude of one vector and the component of the second vector in the direction of the first vector.

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Examples of Scalar Product (i) Power P is the rate of doing work. We know that :- Work W = Force F ∙ Displacement S ∴ = F ∙ Hence, Power P = F ∙ v Thus, power is the scalar product of force and velocity. (ii) The magnetic flux ( Φ ) linked with a plane is defined as scalar product of uniform magnetic field B and vector area A of that plane :- Φ = B ∙ A W t t S

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Properties of Scalar Product (i) The scalar product is commutative. A ∙ B = B ∙ A (ii) The scalar product is distributive. A ∙ (B + C) = A ∙ B + A ∙ C (iii) The scalar product of two mutually perpendicular vectors is zero. (iv) The scalar product of two parallel vectors is equal to the product of their magnitudes . (v) The scalar product of a vector with itself is equal to the square of the magnitude of the vector. (vi) The scalar product of unit orthogonal vectors i, j, k have the following relations :- i ∙ j = j ∙ k = k ∙ i = 0 i ∙ i = j ∙ j = k ∙ k = 1 (vii) The scalar product of two vectors is equal to the sum of the products of their corresponding x-, y-, z- components. A ∙ B = A x B x + A y B y + A z B z ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

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Vector/Cross Product of two Vectors The vector product of two vectors is defined as a vector having a magnitude equal to the product of the magnitudes of the two vectors and the sine of the angle between them, and having the direction perpendicular to the plane containing the two vectors. Thus, if A and B be two vectors, then their vector product, written as A x B, is a vector C defined by :- C = A x B = AB sin (A, B) n, where A and B are the magnitudes of A and B; (A, B) is the angle between them and n is a unit vector perpendicular to the plane of A and B. The direction of C (or n) is perpendicular to the plane containing A and B and its sense is decided by right-hand screw rule. ^ ^ ^

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Examples of Cross Product (i) Suppose there is a particle P of mass m whose position vector is r w.r.t the origin O of an inertial reference frame. Let p (= m v) be the linear momentum of the particle. Then, the angular momentum J of the particle about the origin O is defined as the vector product of r and p, i.e., J = r x p Its scalar magnitude is J = r p sin θ , where θ is the angle between r and p. (ii) The instantaneous linear velocity v of a particle is equal to the vector product of its angular velocity ω and its position vector r with reference to some origin, i.e., v = ω x r Z Y X O θ J r p P

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Properties of Vector Product (i) The vector product is “not” commutative , i.e., A x B ≠ B x A (ii) The vector product is distributive , i.e., A x (B + C) = A x B + A x C (iii) The magnitude of the vector product of two vectors mutually at right angles is equal to the product of the magnitudes of the vectors. (iv) The vector product of two parallel vectors is a null vector (or zero) . (v) The vector product of a vector by itself is a null vector (zero) , i.e., A x A = 0 (vi) The vector product of unit orthogonal vectors i, j, k have the following relations :- (a) i x j = –j x i = k j x k = –k x j = i k x i = –i x k = j (b) i x i = j x j = k x k = 0 (vii) The vector product of two vectors in terms of their x-, y- and z- components can be expressed as a determinant . ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

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THANK YOU Name :- Pankaj Bhootra Class :- 11 C Physics Project