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

2. Vector Calculus

3. 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

4. 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

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

6. 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

7. 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

8. 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

9. 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

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

11. 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

12. Thank You