MATHONGO Formula Sheet - Vector Algebra PDF

MATHONGO – FORMULA SHEET – VECTOR ALGEBRA (a) (i) The position vector of the point which divides the distance between the two end points    ...

MATHONGO – FORMULA SHEET – VECTOR ALGEBRA (a) (i) The position vector of the point which divides the distance between the two end points     mb  na with position vectors a and b in the ratio m: n internally is and externally mn   mb  na is. mn (ii) The position vector of the mid-point of the line joining two points whose position   vectors are a, b is  1   2 ab.  (iii) The position vector of the centroid of the triangle with vertices having position vectors       a b c a, b, c is. 3    (iv) The end points of three vectors a, b and c are collinear if three constant l, m, n (not    all zero) can be found such that la  mb  nc  0 , where l  m  n  0. (b) Components of a vector   (i) If a, b are two non-parallel vectors in a plane, then any vector in that plane can be   represented as la  mb , where l and m are scalars.    (ii) If three constants l, m, n (not all zero) can be found such that la  mb  nc  0 , then    a, b and c are coplanar.     (iii) Any vector d in space can be represented as la  mb  nc , where l, m, n are scalars    and a, b, c are three non coplanar vectors. Addition and subtraction B   If OA  a and AB  b , then the vector   ab    b OB  a  b. O  A If BA is produced to C such that AC  b , then a      b OC  a  b a b C For more such resources and video lecture download the MathonGo App now. http://bit.ly/2CoRyFH MATHONGO – FORMULA SHEET – VECTOR ALGEBRA Vector obey commutative, associative and distributive laws. (i)         m a  b  c  ma  mb  mc (ii)     a b  b a       (iii) a  b  c  a  b  c  a  b  c        Scalar product (or dot product) of two vectors a and b is written as a. b and is defined as       a. b  a b cos , where 0     is the angle between a and b. (a) Scalar multiplication is commutative and distributive              a. b  b.a and a. b  c  a. b  a.c     a.b (b) The orthogonal projection of a on a straight line in the direction of b is given by P   b (c) iˆ. jˆ  1, jˆ. jˆ  1, kˆ. kˆ  1 iˆ. ˆj  0  ˆj. iˆ ˆj. kˆ  0  kˆ. ˆj kˆ. iˆ  0  iˆ. kˆ     (d) a. b  a1b1  a2b2  a3 b3 , where a  a1iˆ  a2 ˆj  a3 kˆ and b  b1iˆ  b2 ˆj  b3 kˆ   a.b a1b1  a2b2  a3 b3 (e) cos     a b a12  a22  a32 b12  b22  b32   (f) Two non-zero vectors are orthogonal if an only if a. b  0       (g) If a. b  0 , then either a  0 or b  0 or a and b are orthogonal.     (h) Work done by a force F in a displacement d is given by F.d. Vector Product (or cross product) For more such resources and video lecture download the MathonGo App now. http://bit.ly/2CoRyFH MATHONGO – FORMULA SHEET – VECTOR ALGEBRA   A vector whose magnitude is equal to the area of the parallelogram witha and b as adjacent     sides and whose direction is perpendicular to both a and b and is such that a , b and that   vector form a right-handed screw is called the vector product of a and b and is denoted by   ab.     Thus a  b = a b sin  n̂ , where  is the angle between the 2 vectors and n̂ is the unit     vector perpendicular to both a and b and a , b and n̂ form a right-handed screw. (a)        ka b  k a  b  a  kb , where k is a scalar.     (b) Cross multiplication by vectors is not commutative a  b  b  a            (c) Cross multiplication is distributive a  b  c  a  b  a  c  (d) iˆ  ˆj  kˆ   ˆj  iˆ ˆj  kˆ  iˆ  kˆ  ˆj kˆ  iˆ  ˆj  iˆ  kˆ iˆ  iˆ  0, ˆj  ˆj  0, kˆ  kˆ  0 iˆ ˆj kˆ     (e) a  b  a1 a2 a3 , where a  a1iˆ  a2 ˆj  a3 kˆ, b  b1iˆ  b2 ˆj  b3 kˆ b1 b2 b3    (f) Area of a triangle whose vertices have position vectors a, b, c is 1       a b  b c  c a 2    (g) Condition for three points with position vectors a, b, c to be collinear is       ab  b c  c a = 0     (h) Moment of a force F about a point P is r  F , where r is the vector of any point on the  line of action of the force F from P. Scalar Triple Product For more such resources and video lecture download the MathonGo App now. http://bit.ly/2CoRyFH MATHONGO – FORMULA SHEET – VECTOR ALGEBRA    If a  a1iˆ  a2 ˆj  a3 kˆ, b  b1iˆ  b2 ˆj  b3 kˆ, c  c1iˆ  c 2 ˆj  c3 kˆ are three vectors, then the dot product of            a and b  c i.e., a. b  c is called the scalar triple product of a, b and c and a1 a2 a3    is denoted by [a b c ] and its value is b1 b2 b3. c1 c 2 c3    (a) The absolute value of the scalar triple product [a b c ] is the volume of the parallelopiped    with a, b, c as its adjacent sides.       (i) If three vectors a, b, c are coplanar, then [a b c ] = 0.     For example: [a b 2a  3b]  0       (ii) If three vectors a, b, c are linearly dependent, then [a b c ] = 0. (b) (i)           a. b  c  a  b.c The positions of dot and cross can be interchanged without affecting the value of the scalar triple product. (ii) If two vectors are interchanged in a scalar triple product, the sign is changed      [a b c ]  [b a c ]  [c b a] (iii) A cyclic permutation of the three vectors does not change the sign or value of the scalar triple product.          [a b c ]  [b c a]  [c a b]           (iv) [a b c  d ]  [a b c ]  [a b d ] Vector Triple Product        If a, b, c are three vectors a  b  c is called a vector triple product. a  b  c   a. c b  a. b c          (i) a  b  c  and a  b  c are different and       (ii) hence placing of the brackets is important. For more such resources and video lecture download the MathonGo App now. http://bit.ly/2CoRyFH MATHONGO – FORMULA SHEET – VECTOR ALGEBRA     (a)                     a  b. c  d  a. c  b. d  a. d b. c =     a.c a.d b.c b.d (b) a  b c  d  [a b d] c  [a b c]d          [a c d ] b  [b c d ] a For more such resources and video lecture download the MathonGo App now. http://bit.ly/2CoRyFH