Rayat Shikshan Sanstha’s S.M.Joshi College, Hadapsar -28 Department of Mathematics Power Point Presentation Topic – Vector Calculus Prof. Darekar S.R

Vector Calculus

Vector-scalar multiplication Vector permits two fundamental operations: add them, multiply them with real number Multiplication gives a vector Has the same direction as that of 2.5v v -v

Vector addition Sum of two vectors Subtraction of two vectors v1-v2 Sum of two vectors Subtraction of two vectors Adding and subtraction of corresponding components of two vectors gives a new vector v2 v1+v2 v1+v2 v1 v1-v2 v1 v1 v2 v2 v2

Linear Combination A linear combination of vectors W = a1v1+a2v2 + a3v3 +…+anvn: all weights are scalars.

Linear Combination of Vectors The combination is Convex if the coefficients sum to 1, and are not negative. Partition of unit v=a1v1 + a2v2 +(1-a1-a2)v3 v3 v2 v=(1-a)v1 + av2 = V1+a(V2-V1) v2 a(V2-V1) v1 v1

Normalize a vector v is represented by n-tuple ( v1,v2,…vn) Magnitude (length): the distance from the tail to the head. Normalization: Scale a vector to have a unity length, unit vector

Dot product Dot product between vector v and vector u gives a scalar If u and v are orthogonal, the dot product equals zero. (a1,a2) dot (b1,b2) = a1xb1 + a2xb2 The most important application of the dot product is to find the angle between two vectors or between two intersecting lines. v u

The Angle between Two Vectors Hence, dot product varies as the cosine of the angle from u to v. v u v v u u v u

Cross Product a = (a1, a2, a3), b=(b1, b2, b3) a x b = ( a2b3 –a3b2), (a3b1-a1b3), (a1b2 – a2b1)

Cross Product v x u u v u x v Cross product between vector v and u gives a vector n is a unit vector perpendicular to both u and v. Follow the right-hand rule u and v are parallel if The length of the cross product equals the area of the parallelogram determined by u and v v x u u v u x v

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