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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 be represent by and . 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, when the inclination is . 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
UNIT VECTOR
A vector of unit magnitude in a direction of vector is called unit vector along and is denoted by a symbolically
EQUAL VECTOR
Two vectors and are said to equal if they have the same magnitude, direction & represent the same physical quantity. This is denoted symbolically by
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 and are collinear
If and only if,
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 :
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
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
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
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
DIRECTIONAL RATIOS:
DIRECTION COSINES OF A LINE:
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:
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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