Assingement Maths Papaer I F.y

7/31/2019 Assingement Maths Papaer I F.y

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Q.1) Define the terms Direction Ratios and Direction Cosine.

Ans. The Any Triple number is proportional to the direction cosines is called as Direction ratios.

It is denoted by a , b, And c.

Direction Cosine:

The Direction Cosine in the the space is ,

X,Y,Z plane and l,m, ,n is respectively.

It is also called dcs.it is denoted by

Q.2) examples of direction Ratios and direction cosine.

DIRECTION RATIOS AND DIRECTION COSINES

If a straight line makes angles

, and

with x-axis , y-axis and z- axis respectively,then cos

,cos ,cos are called direction cosines of the straight line.

Direction cosines are generallydenoted by l , m , n.If O be origin and P is a point whose co-ordinates are (

x , y , z ),then x = lr , y = mr , z = nr , where l , m , n are direction cosines of O

PROOF:

Through P draw PL perpendicular to x-axis so that OL = x.Let OP = rFrom the right triangle OPL , we have

LOP i.e.


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DIRECTION RATIOS OF A LINE:Any three numbers a , b , c proportional to the direction cosines of a line

are called direction ratios of the line.If l , m , n are the direction cosines of a line and are its direction

ratios, thenl =


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Let P( and Q( be two given points,Draw QM and PL perpendiculars to OX and PN perpendicular to

QM.PN = LM = OM

OL =cos

QPN =


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=

, ,

So the direction ratios of the line joining P and Q are


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Q.3) Explain the terms sphere, plane, cylindrical surface.

Spherical Coordinates

* Converting back-and-forth from spherical to Cartesian coordinates

* Phong's model of specular reflection, improved

In the last module we looked at cylindrical coordinates -- a system of coordinates that is very useful

when the important things about a three-dimensional point are its distance from the z-axis and its angle

from the positive xz-plane. In this module we look at situations in which the important things about a

point are its distance from the origin and, using terms from geography, its latitude and longitude. In thissituation we use spherical coordinates.

(rho, phi, theta)

or

Missing equation

* The first of these coordinates -- rho -- denotes the point's distance from the origin. The movie below

shows the sets of points with rho = 0.2, 0.3, ... 1.0.

Missing movie

* The second coordinate -- phi --is very similar to latitude. Think of yourself as located at the origin

with your right hand pointing straight upward along the positive z-axis. Then face the point in question

and lower your right hand until it is pointing at this point. The angle by which your right hand is lowered

is the coordinate phi. Notice if phi = 0 then the point is on the positive z-axis; if phi = pi / 2 then the point


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is in the xy-plane; and if phi = pi then the point is on the negative z-axis. The movie below shows points

with constant values of phi for phi = 0, pi / 16, 2 pi / 16, ... pi.

Missing movie

* The third coordinate -- theta -- is identical to the coordinate theta used in cylindrical coordinates. It

measures the angle from the positive xz-plane to the point. The movie below shows points with

constant values of theta.

Missing movie


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This system of coordinates is very similar to the system -- longitude and latitude -- of coordinates used

to describe points on the earth's surface. You may want to look at the module The Earth is Round --

Most Maps are Flat.

Units

It is important to be aware of the units used with different coordinate systems. In Cartesian coordinates,

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space

where the position of a point is specified by three numbers: the radial distance of that point from a fixed

origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal

projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured

from a fixed reference direction on that plane.

The radial distance is also called the radius or radial coordinate. The polar angle may be called

colatitude, zenith angle, normal angle, or inclination angle.

The use of symbols and the order of the coordinates differs between sources. In one system which is

usual in physics (r, \theta, \varphi) gives the radial distance, polar angle, and azimuthal angle, whereas in

another system used in many mathematics books (r, \theta, \varphi) gives the radial distance, azimuthal

angle, and polar angle. In both systems \rho is often used instead of r. Other conventions are also used

so great care needs to be taken to check which one is being used.

A number of different spherical coordinate systems are used outside mathematics which follow

different conventions. In a geographical coordinate system positions are measured in latitude, longitude

and height or altitude. There are a number of different celestial coordinate systems based on different

fundamental planes and with different terms for the various coordinates. The spherical coordinate

systems used in mathematics normally use radians rather than degrees and measure the azimuthal


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angle counter-clockwise rather than clockwise. The inclination angle is often replaced by the elevation

angle measured from the reference plane.

The concept of spherical coordinates can be extended to higher dimensional spaces and are then

referred to as hyperspherical coordinates.

Assingement Maths Papaer I F.y

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