This document discusses applications of vectors and conics in multivariable calculus. It provides definitions of conic sections including parabolas, circles, ellipses, and hyperbolas. Applications of conics discussed include using the elliptical path of reflection in lithotripsy to break up kidney stones, the design of whispering galleries, and elliptical billiard tables. Additional applications mentioned are the elliptical orbit of planets and the use of parabolic mirrors. Examples of vector applications given are determining landing points while crossing a river, accounting for wind in airplane flight, and the orbital radius vector of planets around the sun.

1. Multivariable Calculus
Topic:
Applications of vectors and conics

2. Group Members
 Shoaib Jalal (DDP-FA13-BSE-098)
 Nauman Shafqat (DDP-FA13-BSE-078)
 Muhammad Nadeem (DDP-FA13-BSE-034)
 Azhar Zafar (DDP-FA13-BSE-016)
 Bilal Zulfiqar (DDP-FA13-BSE-002)
Presented to:
Dr. Sadia Arshad

3. Definition of Conics
 In mathematics, a conic is a curve obtained as the
intersection of a cone (more precisely, a right
circular conical surface) with a plane. In analytic
geometry, a conic may be defined as a plane algebraic
curve of degree 2.
 Types of conic section
1. Parabola
2. Circle and Ellipse
3. Hyperbola

4. Applications of Conics
 The ellipse has an important property that is used in
the reflection of light and sound waves. Any light or
signal that starts at one focus will be reflected to the
other focus. This principle is used in lithotripsy, a
medical procedure for treating kidney stones. The
patient is placed in a elliptical tank of water, with the
kidney stone at one focus. High-energy shock waves
generated at the other focus are concentrated on the
stone, pulverizing it.

5.  The principle is also used in the construction of
"whispering galleries" such as in St. Paul's Cathedral in
London. If a person whispers near one focus, he can
be heard at the other focus, although he cannot be
heard at many places between.

6.  The ability of the ellipse to rebound an object starting
from one focus to the other focus can be
demonstrated with an elliptical billiard table. When a
ball is placed at one focus and is thrust with a cue
stick, it will rebound to the other focus. If the billiard
table is live enough, the ball will continue passing
through each focus and rebound to the other.

7.  One of nature's best known approximations to
parabolas is the path taken by a body projected
upward and obliquely to the pull of gravity, as in the
parabolic trajectory of a golf ball. The friction of air
and the pull of gravity will change slightly the
projectile's path from that of a true parabola, but in
many cases the error is insignificant.

8. Here are some other real life
applications of conic sections
 the paths of the planets around the sun are ellipses with
the sun at one focus
 parabolic mirrors are used to converge light beams at the
focus of the parabola
 parabolic microphones perform a similar function with
sound waves
 solar ovens use parabolic mirrors to converge light beams
to use for heating

9. Definition of Vectors
 A quantity or phenomenon having direction as well as
magnitude, especially as determining the position of
one point in space relative to another.

10. Applications of Vectors
 While crossing a flowing river, you need to know what point
you will land on the opposite bank. This can only be known
with the application of vectors. Here velocity is a vector.

11.  While flying planes in wind, the actual motion of the
plane is along the vector sum of propeller and wind
velocity.

12.  Orbit:
A line connection the Sun and a planet (called
the radius vector) sweeps out equal areas in equal times.