Scalar Product
Scalar / Dot Product of Two Vectors
Product of their
magnitudes
multiplied by the
cosine of the
angle between
the Vectors
Orthogonal Vectors
Angular Dependence
Scalar Product
Scalar Product of a Vector with itself ?
A . A = |A||A| cos 0º
= A2
Scalar Product
Scalar Product of a Vector and Unit vector ?
^x . A =|x||A|cosα
^
= Ax
Yields the component of a vector in a direction of
the unit vector
Where alpha is an angle between A and unit vector
x
Scalar Product
Scalar Product of Rectangular Coordinate
Unit vectors?
x.y = y.z = z.x = ?
=0
x.x = y.y = z.z = ?
=1
Scalar Product Problem 3:
A.B=?
( hint: both vectors have components in
three directions of unit vectors)
Scalar Product Problem 4:
A = y3 + z2; B= x5 + y8
A.B=?
Scalar Product Problem 5:
A = -x7 + y12 +z3;
B = x4 + y2 + z16
A.B = ?
Line Integrals
Line Integrals
Line Integrals
Line Integrals
Line Integrals
Line Integrals
Line Integrals
Line Integrals
Spherical coordinates
Spherical coordinates
Spherical Coordinates
For many mathematical problems,
it is far easier to use spherical
coordinates instead of Cartesian
ones. In essence, a vector r (we
drop the underlining here) with the
Cartesian coordinates (x,y,z) is
expressed in spherical coordinates
by giving its distance from the
origin (assumed to be identical for
both systems) |r|, and the two
angles  and  between the
direction of r and the x- and z-axis
of the Cartesian system. This
sounds more complicated than it
actually is:  and  are nothing
but the geographic longitude
and latitude. The picture below
illustrates this
Spherical coordinate system
Simulation of SCS
• http://www.flashandmath.com/mathlets/mu
lticalc/coords/index.html
Line Integrals
Line Integrals
Line Integrals
Line Integrals
Tutorial
• Evaluate:
4
xy
ò ds
c
Where C is right half of the circle : x2+y2=16
Solution
We first need a parameterization of the circle. This is given by,
We now need a range of t’s that will give the right half of the circle. The
following range of t’s will do this:
Now, we need the derivatives of the parametric equations and let’s
compute ds:
Tutorial ………
• The line integral is then :
Assignment No 3
• Q. No. 1: Evaluate
curve shown below.
where C is the
Assignment No 3: ….
• Q.NO 2: Evaluate
were C is the line segment
from
to