View by Category

The presentation will start after a short

(15 second) video ad from one of our sponsors.

Hot tip: Video ads won’t appear to registered users who are logged in. And it’s free to register and free to log in!

(15 second) video ad from one of our sponsors.

Hot tip: Video ads won’t appear to registered users who are logged in. And it’s free to register and free to log in!

Loading...

PPT – Basics of Analytical Geometry PowerPoint presentation | free to download - id: f502b-ZDc1Z

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Basics of Analytical Geometry

- By
- Kishore Kulkarni

Outline

- 2D Geometry
- Straight Lines, Pair of Straight Lines
- Conic Sections
- Circles, Ellipse, Parabola, Hyperbola
- 3D Geometry
- Straight Lines, Planes, Sphere, Cylinders
- Vectors
- 2D 3D Position Vectors
- Dot Product, Cross Product Box Product
- Analogy between Scalar and vector representations

2D Geometry

- Straight Line
- ax by c 0
- y mx c, m is slope and c is the y-intercept.
- Pair of Straight Lines
- ax2 by2 2hxy 2gx 2fy c 0
- where abc 2fgh af2 bg2 ch2 0

Conic Sections

- Circle, Parabola, Ellipse, Hyperbola
- Circle Section Parallel to the base of the cone
- Parabola - Section inclined to the base of the

cone and intersecting the base of the cone - Ellipse - Section inclined to the base of the

cone and not intersecting the base of the cone - Hyperbola Section Perpendicular to the base of

the cone

Conic Sections

- Circle x2 y2 r2 , r gt radius of circle
- Parabola y2 4ax or x2 4ay
- Ellipse x2/a2 y2/b2 1, a is major axis b

is minor axis - Hyperbola x2/a2 - y2/b2 1.
- In all the above equation, center is the origin.
- Replacing x by x-h and y by y-k, we get equations

- with center (h,k)

Conic Sections

- In general, any conic section is given by
- ax2 by2 2hxy 2gx 2fy c 0
- where abc 2fgh af2 bg2 ch2 ! 0
- Special cases
- h2 ab, it is a parabola
- h2 lt ab, it is an ellipse
- h2 gt ab, it is a hyperbola
- h2 lt ab and ab, it is a circle

3D Geometry

- Plane - ax by cz d 0
- Sphere - x2 y2 z2 r2
- (x-h)2 (y-k)2 (z-l)2 r2 , if center is (h,

k, l) - Cylinder - x2 y2 r2, r is radius of the base.

- (x-h)2 (y-k)2 r2 , if center is (h, k, l)

3D Geometry

- Question
- What region does this inequality represent in a

3D space ?

9 lt x2 y2 z2 lt 25

3D Geometry

- Straight Lines
- Parametric equations of line passing through (x0,

y0, z0) - x x0 at, y y0 bt, z z0 ct
- Symmetric form of line passing through (x0, y0,

z0) - (x - x0)/a (y - y0)/b (z - z0)/c
- where a, b, c are the direction numbers of the

line.

Vectors

- Any point in P in a 2D plane or 3D space can be

represented by a position vector OP, where O is

the origin. - Hence P(a,b) in 2D corresponds to position vector

lt a, bgt and Q(a, b, c) in 3D space corresponds to

position vector lt a, b, cgt - Let P ltx1, y1, z1gt and Q lt x2, y2, z2 gt then

vector PQ OQ OP lt x2 x1, y2 y1, z2

z1gt - Length of a vector v lt v1, v2, v3gt is given by
- v sqrt(v12 v22 v32)

Dot (Scalar) Product of vectors

- Dot product of two vectors a a1i a2j a3k
- and b b1i b2j b3k is defined as
- a.b a1b1 a2b2 a3b3.
- Dot Product of two vectors is a scalar.
- If ? is the angle between a and b, we can write
- a.b abcos?
- Hence a.b 0 implies two vectors are orthogonal.

- Further a.b gt 0 we can say that they are in the

same general direction and a.b lt 0 they are in

the opposite general direction. - Projection of vector b on a a.b / a
- Vector Projection of vector b on a (a.b / a)

( a / a)

Direction Angles and Direction Cosines

- Direction Angles a, ß, ? of a vector a a1i

a2j a3k are the angles made by a with the

positive directions of x, y, z axes respectively. - Direction cosines are the cosines of these

angles. We have - cos a a1/ a, cos ß a2/ a, cos ? a3/

a. - Hence cos2 a cos2 ß cos2 ? 1.
- Vector a a ltcos a, cos ß, cos ?gt

Cross (Vector) Product of vectors

- Cross product of two vectors a a1i a2j a3k
- and b b1i b2j b3k is defined as
- a x b (a2b3 a3b2)i (a3b1 a1b3)j (a1b2

a2b1)k. - a x b is a vector.
- a x b is perpendicular to both a and b.
- a x b a b sin? represents area of

parallelogram.

Cross (Vector) Product

- Question
- What can you say about the cross product of

two vectors in 2D ?

Box Product of vectors

- Box Product of vectors a, b and c is defined as
- V a.(b x c)
- Box Product is also called Scalar Tripple Product
- Box product gives the volume of a parallelepiped.

Vector Equations

- Equation of a line L with a point P(x0, y0, z0)

is given by - r r0 tv
- where r0 lt x0, y0, z0gt, r lt x, y, zgt, v

lta, b, cgt is a vector parallel to L, t is a

scalar. - Equation of a plane is given by
- n.(r - r0) 0
- where n is a normal vector, which is analogous

to the scalar equation - a (x- x0) b (y- y0) c (z- z0) 0

Vector Equations

- Let a and b be position vectors of points
- A(x1, y1,z1) and B(x2, y2,z2). Then position

vector of the point P dividing the vector AB in

the ratio mn is given by - p (mb na) / (mn)
- which corresponds to
- P ((mx2 nx1)/(mn), (my2 ny1)/(mn), (mz2

nz1)/(mn))

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Page of

CrystalGraphics Sales Tel: (800) 394-0700 x 1 or Send an email

Home About Us Terms and Conditions Privacy Policy Contact Us Send Us Feedback

Copyright 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Basics of Analytical Geometry" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!

Committed to assisting Temple University and other schools with their online training by sharing educational presentations for free