Basic Maths Vectors

VECTORS A vector has both magnitude and sense of direction, and follows triangle law of vector addition. For example, displacement, velocity, and force are vectors. r r Vector quantities are usually denoted by putting an arrow over the corresponding letter, as A or a . Sometimes in print work (books) vector quantities are usually denoted by boldface letters as A or a. r r Magnitude of a vector A is a positive scalar and written as A or A. Unit Vector It is mathematical way to express direction of a vector and defined by the ratio of a vector to its magnitude. When a unit vector is multiplied with a scalar magnitude, we get a vector of corresponding magnitude in the direction of the unit vector. A unit vector is usually represented by putting a sign (ˆ) known as cap, hat or caret over a letter assigned to the unit vector. This letter may be the same as used for the vector, or its lower case letter, or some other r symbol. For example, if we assign lower case letter a to unit vector in the direction of vector A , the unit vector denoted by r A  Aaˆ â is expressed by the following equation. Geometrical Representation of Vectors. Geometrically a vector is represented by a directed straight-line segment drawn to a scale. Starting point of the directed line segment is known as tail and the end-point as arrow, head, or tip. The orientation of the line and the arrow collectively show the direction and the length of the line drawn to a scale shows the magnitude. For example let a particle moves from point A to B following a curvilinear path shown in the figure. It displacement vector is straight line AB directed form A to B. If straight-line distance between A and B is 25 m, the directed line segment has to be drawn to suitable scale. If we assume the scale 1.0 cm = 10 m, the geometrical length of the displacement vector AB must be 2.5 cm. Arrow B B Displacement A Addition of Vectors: The Triangle Law Tail Path A Geometrical representation of Displacement Vector Use of geometry in solving problems involving vectors is of fundamental nature. The triangle law also uses principles of plane geometry. This law states: The vectors to be added are drawn in such a manner that the tail of a vector coincides the tip of the preceding vector (in tip to tail fashion); their resultant is defined by the vector drawn from the tail of the first vector to the tip of the second vector. The two vectors to be added and their resultant are coplanar. r r Consider vectors A and B shown in the figure-I. Using the triangle law, we obtain geometrical construction r r r shown in the figure-II, where it is shown that two vectors and their sum C  A  B always make a closed triangle. BASIC MATHS & VECTOR r r r r r If we change order of vector A and B , it shown in figure-III that sum given by equation C  B  A remain unchanged. Therefore, vector addition is commutative. A A C B B B C A Fig. (i) Fig. (ii) Fig. (iii) Construction, which is combination of the figure-II and III, is in form of a parallelogram and is shown in figure-IV. Geometry of the above figure suggests the following results. B sin   1  C  A2  B2  2 AB cos  ;   tan  A  B cos   Ex. Sol. r r A vector A and B make angles of 20° and 110° respectively with the X–axis. The magnitudes of these vectors are 5m and 12m respectively. Find their resultant vector. r r Angle between the A and B = 110°– 20° = 90° R R  A2  B2  2 AB cos 90 o  5 2  12 2  13m B r r Let angle of R from A is  tan   B sin  1 2 sin 90 o 12  1 12    o A  B cos  5  12 cos 90 5  12  0 5 A ° 110 20° X-axis r 12  or   tan 1   with vector A or (+ 20°) with X–axis  5  Two forces each numerically equal to 10 dynes are acting as shown in the figure, then find resultant of these two vectors. Sol. The angle  between the two vectors is 120° and not 60°. = 1 0 0  1 0 0  1 0 0  1 0 dyne 10  R  (10 )2  (1 0 )2  2(10 )(1 0 )(cos1 2 0 o ) dy ne Ex. 60° 10 dyne Addition of more than two Vectors The triangle law can be extended to define addition of more than two vectors. Accordingly, if vectors to be added are drawn in tip to tail fashion, resultant is defined by a vector drawn from the tail of the first vector to the tip of the last vector. This is also known as the polygon rule for vector addition. r r r r Operation of addition of three vectors A, B and C and their resultant P are shown in figure. C P r r r r ABC P B A Here it is not necessary that three or more vectors and their resultant are coplanar. In fact, the vectors to be added and their resultant may be in different planes. However if all the vectors to be added are coplanar, their resultant must also be in the same plane containing the vectors. Subtraction of Vectors r r A vector opposite in direction but equal in magnitude to another vector A is known as negative vector of A . It r is written as – A . Addition of a vector and its negative vector results a vector of zero magnitude, which is known r as a null vector. A null vector is denoted by arrowed zero  0  . The idea of negative vector explains operation of subtraction as addition of negative vector. Accordingly to r r r r subtract a vector from another consider vectors A and B shown in the figure. To subtract B from A , the r r negative vector – B is added to A according to the triangle law as shown in figure-II. A A B A B B Multiplying by a number Multiplication by a positive number changes magnitude of the vector but not the direction and multiplication by a negative number changes magnitude and reverses direction. r Thus multiplying a vector by a number n makes magnitude of the vector n times. nA   nA  aˆ r Here â denotes the unit vector in the direction of vector A . Resolution of a Vector into Components Following laws of vector addition, a vector can be represented as a sum of two (in two-dimensional space) or three (in three-dimensional space) vectors each along predetermined directions. These directions are called axes and parts of the original vector along these axes are called components of the vector. Cartesian components in two dimensions If a vector is resolved into its components along mutually perpendicular directions, the components are called Cartesian or rectangular components. r r r In figure is shown, a vector A resolved into its Cartesian components A x and A y along the x and y-axis. Magnitudes Ax and Ay of these components are given by the following equation. Ax=Acos and Ay = Asin r A  A xˆi  A yˆj A  A x2  A y2 BASIC MATHS & VECTOR Here î and ˆj are the unit vectors for x and y coordinates respectively.. Mathematical operations e.g. addition, subtraction, differentiation and integration can be performed independently on these components. This is why in most of the problems use of Cartesian components becomes desirable. Cartesian components in three dimensions r A vector A resolved into its three Cartesian components one along each of the directions x, y, and z-axis is shown in the figure. z A AX Equal Vectors x AZ r r r r y A  A x  A y  A z  A xˆi  A yˆj  A z kˆ ; A  A 2x  A 2y  A 2z Ay Two vectors of equal magnitudes and same directions are known as equal vectors. Their x, y and z components in the same coordinates system must be equal. r r If two vectors a  a xˆi  a yˆj  a z kˆ and b  b xˆi  b yˆj  b z kˆ are equal vectors, we have r r a  b  ax=bx, ay=by and az=bz Parallel Vectors Two parallel vectors must have the same direction and may have unequal magnitudes. Their x, y and z components in the same coordinate system bear the same ratio. r r Consider two vectors a  a xˆi  a yˆj  a z kˆ and b  b xˆi  b yˆj  b z kˆ , if they are parallel, we have ay az a r r a b  x   bx by bz Product of Vectors (i) In all physical situation, whose description involve product of two vectors, only two categories are observed. One category where product is also a vector involves multiplication of magnitudes of two vectors and sine of the angle between them, while the other category where product is a scalar involves multiplication of magnitudes of two vectors and cosine of the angle between them. Accordingly, we define two kinds of product operation. The former category is known as vector or cross product and the latter category as scalar or dot product. Scalar or dot product of two vectors r r The scalar product of two vectors A and B equals to the product of their magnitudes and the cosine of the angle between them. r r A  B  AB cos   OA  OB  cos  The above equation can also be written in the following ways. r r A  B   A cos   B  OP  OB r r A  B  A  B cos    OA  OQ Above two equations and figures, suggest a scalar product as product of magnitude of the one vector and magnitude of the component of another vector in the direction of the former vector. (i) (ii) (iii) (iv) (v) r r r r Dot product of two vectors is commutative: A  B  B  A r r r r If two vectors are perpendicular, their dot product is zero. A  B  0 , if A  B r r r r Dot product of a vector by itself is known as self-product. A  A  A 2  A  A  A r r  A  B 1 The angle between the vectors   cos   AB  A^ r r A (A) Component of A in direction of B r r r r r r r  A.B  r  ˆ A.B ˆ  A r r B ˆ ˆ ˆ A| |   A cos   B     r  B   A.B  B  A B  B   B r r r r r (B) Component of A perpendicular to B : A   A  A II (vi) (vii) Ex. Sol. Dot product of Cartesian unit vectors: r r If A  A xˆi  A yˆj  A z kˆ and B  B xˆi  B yˆj  B z kˆ , their dot product is given by r r A  B  A xBx  A yB y  A zB z r r r r r r If | A  B|  | A  B| , then find the angle between A and B . r r r r   A2  | A  B|  | A  B| or Ex. Sol. ˆi  ˆi  ˆj  ˆj  kˆ  kˆ  1 ˆi  ˆj  ˆj  kˆ  kˆ  ˆi  0 A2 + B2 + 2AB cos  = A2 + B2 – 2AB cos  or A2  cos  = 0 r r r r If A  4 ˆi  nˆj  2kˆ and B  2 ˆi  3 ˆj  kˆ , then find the value of n so that A  B r r Dot product of two mutually perpendicular vectors is zero A  B  0  (4iˆ  njˆ  2kˆ).(2iˆ  3 ˆj  kˆ)  0  (4  2)  (n  3)  (2 1)  0  3n = – 6    = 90° A AII BASIC MATHS & VECTOR (ii) Vector or cross product of two vectors r r r The vector product C of two vectors A and B is defined as r r (A) Its magnitude is the product of magnitudes of A and B and of the sine of angle  between vectors r r A and B . r r (B) Its direction is perpendicular to the plane containing vectors A and B and is decided by right hand rule by curling fingers in the direction from the first vector towards the second vector. In figure, where it is represented by n̂ . r r r C  A  B   AB sin   nˆ On paper vectors perpendicularly out and into the plane of paper are represented encircled dot e and encircled r r by r cross  signs respectively. Following this convention, cross product C  A  B is shown in the figure. To have different symbols for scalar and vector products, symbols dot (  ) and cross (×) respectively are written between the vectors undergoing these operations. r r r Cross product C  A  B , can also be written in the following ways. r r r r r r C  A  B  A  B sin   nˆ C  A  B   A sin   Bnˆ The above two equations and figures explain that the magnitude of vector or cross product is the product of magnitude of one vector and magnitude of the component of the other vector in the direction perpendicular to the first one. Ex. Sol. ˆ and (iˆ  ˆj  2 k) ˆ . Find a unit vector perpendicular to both the vectors (2 ˆi  3 ˆj  k) r r Let A  2 ˆi  3 ˆj  kˆ and B  ˆi  ˆj  2kˆ r r r r A B unit vector perpendicular to both A and B is nˆ  r r A B iˆ r r A B  2 ˆj 3 kˆ 1  iˆ(6  1)  ˆj (4  1)  kˆ( 2  3)  7iˆ  3 ˆj  5kˆ 1 1 2 r r  A  B  7 2  ( 3) 2  ( 5) 2  83 unit  nˆ  1 (7iˆ  3 ˆj  5k ) 83 (i) Vector product of two vectors is always a vector perpendicular to the plane containing the two vectors, i.e., orthogonal r r (perpendicular) to both the vectors A and B . r r r r AB Unit vector perpendicular to A and B is n̂   r r | A  B| A× B= C (ii) Vector product of two vectors is not commutative i.e. cross products B B r r r r A A A  B and B  A have equal magnitudes but opposite directions as B × A = -C shown in the figure. r r r r A  B  B  A (iii)The vector product is distributive when the order of the vectors is strictly maintained, r r r r r r r i.e. A  (B  C)  A  B  A  C r r r r | A  B|  1 r  (iv) Angle between two vectors A and B is given by   sin  r | A| | B|  (v) The self cross product, i.e., product of a vector by itself is a zero vector or a null vector.  r r r A  A  ( AA sin 0) nˆ  0 = ˆi  ˆi  ˆj  ˆj  kˆ  kˆ (vi) In case of orthogonal unit vectors ˆi, ˆj and kˆ ; according to right hand thumb rule ˆ ˆj  kˆ  ˆi, kˆ  ˆi  ˆj and ˆj  ˆi  k, ˆ kˆ  ˆj  ˆi, ˆi  kˆ  ˆj ˆi  ˆj  k, r r (v) If A  A xˆi  A yˆj  A z kˆ and B  B xˆi  B yˆj  B z kˆ , their cross-products is given by ˆi r r A  B  Ax ˆj Ay kˆ ˆ A z  ˆi(A y B z  A z B y )  ˆj(A x B z  A z B x )  k(A xBy  AyBx ) Bx By Bz r r r r r r (vi) If A , B and C are coplanar, then A.(B  C)  0 . Rate of change of a vector with time It is derivative of a vector function with respect to time. Cartesian components of a time dependent vector, if r given as function of time as r  t   x(t)iˆ  y(t)jˆ  z(t)kˆ , the time rate of change can be calculated according to equation r dr  t  dx(t )iˆ dy (t ) ˆj dz (t )kˆ    dt dt dt dt Methods of differentiation of vector functions Methods of differentiation of scalar functions are also applicable to differentiation of vector functions. r r r r d r r dF dG d r r dF r r dG      F  G   F G G F i. ii. dt dt dt r dt dt dt r d dX r dF XF  FX iii. Here X is a scalar function of time. dt dt r dt r r r d r r dF r r dG F G  G  F  iv. Order of the vector functions F and G dt dt dt        