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Three Dimensional Geometry

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Three Dimensional Geometry PDF Notes, Important Questions and Formulas

 

Vector

 

 

KINDS OF VECTORS

 

DEFINITIONS:

 

A vector may be described as a quantity having both magnitude & direction. A vector is generally represented by a directed line segment, say AB. A is called the initial point & B is called the terminal point. The magnitude of vector

AB is expressed by | AB |

 

The modulus, or magnitude, of a vector is the positive number which is the measure of its length. The modulus of the vector a is sometimes denoted by |a|, and sometimes by the corresponding symbol a in italics. The vector which has the same modulus as a, but the opposite direction, is called the negative of "a", and is denoted by –a.

 

Let the vectors begin mathsize 12px style straight a with rightwards arrow on top comma straight b with rightwards arrow on top end style be represent by begin mathsize 12px style OA with rightwards arrow on top end style and begin mathsize 12px style OB with rightwards arrow on top end style. Then the inclination of the vectors, or the angle between them, is defined as that angle AOB which does not exceed π.  Thus if θ denote this inclination, begin mathsize 12px style 0 less or equal than straight theta less or equal than straight pi end style when the inclination is begin mathsize 12px style straight pi divided by 2 end style. The vector are said to be perpendicular, when it is 0 or π they are parallel.

ZERO VECTOR

A vector of zero magnitude i.e. which has the same initial & terminal point, is called a Zero Vector. It is denoted by begin mathsize 12px style 0 with rightwards arrow on top end style

UNIT VECTOR

A vector of unit magnitude in a direction of begin mathsize 12px style straight a with rightwards arrow on top end style vector begin mathsize 12px style straight a with rightwards arrow on top end style is called unit vector along begin mathsize 12px style straight a with rightwards arrow on top end style and is denoted by a symbolically

begin mathsize 12px style straight a with hat on top equals fraction numerator straight a with rightwards arrow on top over denominator vertical line straight a with rightwards arrow on top vertical line end fraction end style

EQUAL VECTOR

Two vectors begin mathsize 12px style straight a with rightwards arrow on top end style and begin mathsize 12px style straight b with rightwards arrow on top end style are said to equal if they have the same magnitude, direction & represent the same physical quantity. This is denoted symbolically by begin mathsize 12px style straight a with rightwards arrow on top equals straight b with rightwards arrow on top end style

COLLINEAR VECTORS

 

Two vectors are said to be collinear if their directed line segments are parallel disregards to their direction. Collinear

vectors are also called Parallel Vectors. If they have the same direction they are named as like vectors otherwise unlike vectors.

Symbolically, two non zero vectors begin mathsize 12px style straight a with rightwards arrow on top end style and begin mathsize 12px style straight b with rightwards arrow on top end style are collinear

If and only if, begin mathsize 12px style straight a with rightwards arrow on top equals straight K straight b with rightwards arrow on top comma space where space straight K element of straight R end style

COPLANAR VECTORS

A given number of vectors are called coplanar if their directed line segments are all parallel to the same plane. Note that "Two Vectors Are Always Coplanar".

 

Remark

Vectors as defined above are usually called free vectors, since the value of such a vector depends only on its length and direction and is independent of its position in space. A single free vector cannot therefore completely represent the effect of a localized vector quantity, such as a force acting on a rigid body. This effect depends on the line of action of the force; and it will be shown later that two free vectors are necessary for its specification.

 

ADDITION AND SUBTRACTION OF VECTORS

Triangle Law

The manner in which the vector quantities of mechanics and physics are compounded is expressed by the triangle law of addition, which may be stated as follows :

begin mathsize 12px style table attributes columnalign left end attributes row cell If text  three points O,Q,R are end text end cell row cell text chosen so that  end text OP with rightwards arrow on top equals straight a with rightwards arrow on top and end cell row cell PR with rightwards arrow on top equals straight b with rightwards arrow on top then end cell row cell The text  vector  end text OR with rightwards arrow on top text  is called end text end cell row cell text the (vector) sum or resultant end text end cell row cell text of  end text straight a with rightwards arrow on top text  and  end text straight b with rightwards arrow on top end cell end table end style

 

 

 Dimensions Geometry

 

DISTANCE BETWEEN TWO POINTS

 

Let P and Q be two given points in space. Let the co-ordinates of the points P and Q be (x1, y1 z1) and (x2, y2, z2) with respect to a set OX, OY, OZ of rectangular axes. The position vectors of the points P

 

 begin mathsize 12px style table attributes columnalign left end attributes row cell text and Q are given by  end text OP with rightwards arrow on top equals straight x subscript 1 straight i with hat on top plus straight y subscript 1 straight j with hat on top plus straight z subscript 1 straight k with hat on top text  and end text end cell row cell OQ with rightwards arrow on top equals straight x subscript 2 straight i with hat on top plus straight y subscript 2 straight j with hat on top plus straight z subscript 2 straight k with hat on top end cell row cell Now text  we have  end text PQ with rightwards arrow on top equals OQ with rightwards arrow on top minus OP with rightwards arrow on top. end cell row cell equals left parenthesis straight x subscript 2 straight i with hat on top plus straight y subscript 2 straight j with hat on top plus straight z subscript 2 straight k with hat on top right parenthesis minus left parenthesis straight x subscript 2 straight i with hat on top plus straight y subscript 2 straight j with hat on top plus straight z subscript 2 straight k with hat on top right parenthesis end cell row cell equals left parenthesis straight x subscript 2 minus straight x subscript 1 right parenthesis straight i with hat on top minus left parenthesis straight y subscript 2 minus straight y subscript 1 right parenthesis straight j with hat on top minus left parenthesis straight z subscript 2 minus straight z subscript 1 right parenthesis straight k with hat on top end cell row cell PQ equals vertical line PQ with rightwards arrow on top vertical line equals square root of left parenthesis straight x subscript 2 minus straight x subscript 1 right parenthesis squared minus left parenthesis straight y subscript 2 minus straight y subscript 1 right parenthesis squared minus left parenthesis straight z subscript 2 minus straight z subscript 1 right parenthesis squared end root end cell row cell Distance left parenthesis straight d right parenthesis text  between two points (x end text subscript 1 comma end subscript straight y subscript 1 comma end subscript straight z subscript 1 right parenthesis text   end text and text  (x end text subscript 2 comma end subscript straight y subscript 2 comma end subscript straight z subscript 2 right parenthesis text  is  end text end cell row cell text d= end text square root of left parenthesis straight x subscript 2 minus straight x subscript 1 right parenthesis squared minus left parenthesis straight y subscript 2 minus straight y subscript 1 right parenthesis squared minus left parenthesis straight z subscript 2 minus straight z subscript 1 right parenthesis squared end root end cell end table end style

 

 

CENTROID OF A TRIANGLE:

 

Let ABC be a triangle. Let the co-ordinates of the vertices A, B and C be (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) respectively. Let AD be a median of the ABC. Thus D is the mid point of BC.  The co-ordinates of D are

 

 begin mathsize 12px style open parentheses fraction numerator begin display style straight x subscript 2 plus straight x subscript 3 end style over denominator begin display style 2 end style end fraction comma fraction numerator begin display style straight y subscript 2 plus straight y subscript 3 end style over denominator begin display style 2 end style end fraction comma fraction numerator begin display style straight z subscript 2 plus straight z subscript 3 end style over denominator begin display style 2 end style end fraction close parentheses end style

 

CENTROID OF A TETRAHEDRON:

Let ABCD be a tetrahedron, the co-ordinates of whose vertices are (xr, yr, zr), r = 1, 2, 3, 4. Let G1 be the centroid of the face ABC of the tetrahedron. Then the co-ordinates of G1 are

 begin mathsize 12px style open parentheses fraction numerator begin display style straight x subscript 1 plus straight x subscript 2 plus straight x subscript 3 end style over denominator begin display style 3 end style end fraction comma fraction numerator begin display style straight y subscript 1 plus straight y subscript 2 plus straight y subscript 3 end style over denominator begin display style 3 end style end fraction comma fraction numerator begin display style straight z subscript 1 plus straight z subscript 2 plus straight z subscript 3 end style over denominator begin display style 3 end style end fraction close parentheses end style

 

The fourth vertex D of the tetrahedron does not lie in the plane of ABC. We know from statics that the centroid of the tetrahedron divides the line DG1 in the ratio 3:1. Let G be the centroid of the tetrahedron and if (x, y, z) are its co-ordinates, then

 begin mathsize 12px style table attributes columnalign left end attributes row cell straight x equals fraction numerator 3. fraction numerator straight x subscript 1 plus straight x subscript 2 plus straight x subscript 3 over denominator 3 end fraction plus 1. straight x subscript 4 over denominator 3 plus 1 end fraction or text  x= end text fraction numerator straight x subscript 1 plus straight x subscript 2 plus straight x subscript 3 plus straight x subscript 4 over denominator 4 end fraction end cell row similarly row cell straight y equals 1 fourth left parenthesis straight y subscript 1 plus straight y subscript 2 plus straight y subscript 3 plus straight y subscript 4 right parenthesis comma text  z= end text 1 fourth left parenthesis straight z subscript 1 plus straight z subscript 2 plus straight z subscript 3 plus straight z subscript 4 right parenthesis end cell end table end style

 

DIRECTIONAL RATIOS:

 

DIRECTION COSINES OF A LINE:

 

 begin mathsize 12px style table attributes columnalign left end attributes row cell If text   end text straight alpha text ,  end text straight beta text ,  end text straight chi text  are the angles which a given directed line end text end cell row cell makes text  with the positive directions of the axes. ofx,y end text end cell row cell text and z respectively, then cos  end text straight alpha text , cos  end text straight beta text  cos  end text straight chi text  are called end text end cell row cell text the direction cosines(briefly written as d.c.'s) of the  end text end cell row cell line. text  These d.c.s' are usually denote by l, m, n. end text end cell row cell text Let AB be a given line. Draw a line OP parallel to the end text end cell row cell text line AB and passing through the origin O. Measure end text end cell row blank row cell text angles  end text straight alpha text ,  end text straight beta text ,  end text straight chi comma text  then cos  end text straight alpha comma cos text   end text straight beta text ,cos  end text straight chi text  are the d.c.'s of end text end cell row cell text the line AB. It can be easily seen that l, m, n, are the end text end cell row cell text direction cosines of a line if and only if l end text straight i with hat on top plus straight m straight j with hat on top plus straight n straight k with hat on top end cell row cell is text  a unit vector in the direction of that line. end text end cell row blank end table end style

 

 

 Dimensions Geometry

 

SAMPLE SPACE DEFINITION, ODDS IN FAVOR, PERMUTATION & COMBINATION The theory of probability originated from the game of chance and gambling. In days of old, gamblers used to gamble in a gambling house with a die to win the amount fixed among themselves. They were always desirous to get the prescribed number on the upper face of a die when it was thrown on a board. Shakuni of Mahabharat was perhaps one of them. People started to study the subject of probability from the middle of seventeenth century. The mathematicians Huygens, Pascal Fermat and Bernoulli contributed a lot to this branch of Mathematics. A.N. Kolmogorow proposed the set theoretic model to the theory of probability.

 

DEFINITIONS:

 

  1. Experiment: An action or operation resulting in two or more outcome.
  2. Sample space : A set S that consists of all possible outcomes of a random experiment is called a sample space and each outcome is called a sample point often there will be more than one sample space that can describes outcomes of an experiment, but there is usually only one that will provide the most information. If a sample space has a finite number of points it is called finite sample space and infinite sample space if it has infinite number of points.

 

CLASSICAL DEFINITION OF PROBABILITY:

If n represents the total number of equally likely, mutually exclusive and exhaustive outcomes of an experiment and m of them are favourable to the happening of the event A, then the probability of happening of the event A is given by P(A) = m/n.

 

 

 

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left parenthesis s e c squared space a right parenthesis i with rightwards harpoon with barb upwards on top plus j with rightwards harpoon with barb upwards on top plus k with rightwards harpoon with barb upwards on top comma space i with rightwards harpoon with barb upwards on top plus left parenthesis s e c squared b right parenthesis j with rightwards harpoon with barb upwards on top plus k with rightwards harpoon with barb upwards on top space a n d space i with rightwards harpoon with barb upwards on top plus j with rightwards harpoon with barb upwards on top plus left parenthesis s e c squared c right parenthesis k with rightwards harpoon with barb upwards on top space a r e space c o p l a n a r.
w h a t space i s space t h e space v a l u e space o f space cos e c to the power of 2 space end exponent a plus cos e c squared b plus cos e c squared c ?

28 January 2019 11:01 AM

Sir pls solve the following.

7 January 2019 10:34 AM
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