( PRE – FINAL DRAFT / TO CONTINUE )
C O N T E N T S >
- Unit vector
- Basic set
- Vector addition > Triangular / Parallelogram / Polygonal / Analytical
- Vector subtraction
- Lami’s theorem
- Directional Cosines
- Vector Multiplication > Vector / Cross product … Determinant & Component Methods
- Scalar / Dot product
- Vector components
- Method of components
- Vector classification
- Scalar triple / mixed triple / boxed product.
Physical quantities are divided into vectors and scalars. A scalar quantity is denoted by a magnitude or real number-Eg.Temperature of a room,volume of a jug,density of steel or presssure of air in a tyre.In the case of vector quantities,a specific direction related to some underlying reference frame is needed to define it,in addition to a magnitude…Eg.displacement of a car and velocity of a body.The length of a vector is called it’s magnitude.It is indicated by using vertical bars or by using italics – | A| or A .If the length of a vector is one unit,then it’s a unit vector.
A component of a vector is the projection of the vector on an axis.The find this,we draw perpendicular lines from from the two ends of the vector to the axis.The process of finding the components is called resolving the vector.The component of a vector has the same direction [ along an axis ] as the vector.
A unit vector lacks both dimension and unit,it only specifies a direction.Regarding the relations among vectors,we have great freedom in choosing a co-ordinate system,as the relations does not depend on the location of the origin of the co-ordinate system or on the orientation of the axis.
Basic Set : Consider two non zero vectors, a and b,where the direction of b is neither the same or the opposite to that of a .Let OA and OB be representations of a and b and P is the plane of the triangle, OAB.Now,any vector v whose representation OV lies in the plane P can be written as v = λ a + μ b . Here the co-efficients, λ and μ are unique.As the vectors have their directions parellel to the same plane,they are coplanar.Any vector coplanar with a and b can be expanded uniquely in the above form.Also, the expansion set cannot be reduced in number[ say,to a single vector ].Hence the pair of vectors ( a,b) is said to be a basic set for vectors lying in the plane P.If we are dealing with three co-planar vectrors,a,b and c , then it’s v = λ a + μ b + ν c , again a,b,c is a basic set..Eventhough any set of three non-coplanar vectors form a basic,the basic vectors are considered as orthogonal unit vectors. Here, the basic set is denoted by ( i,j,k ).
Position vector : Vectors which are used to specify the positions of points in space.
A vector does not necessarily have location,eventhough a vector may refer to a quantity defined at a particular point.Two vectors can be compared ,eventhough they measure a physical quantity defined at different points of space & time.Note- the applicability of vectors is largely based on euclidean geometry…. that space is flat [ for huge distances ].In such a scanario, we can compare two vectors at differtent points.
A Vector must a) satisfy the parallelogram law of addition b) have a magnitude and direction independent of choice of co-ordinate system.
All quantities having magnitude and direction need not be vectors – For eg, in the case of the rotation of a rigid body about a particular axis fixed in space, eventhough it has both magnitude ( angle of rotation ) and direction ( the direction of the axis ), but two such rotations do not combine according to the vector law of addition, unless the angles of rotation are infinitesimally small. Eg – when the two axes are perpendicular to each other and the rotations are by π/2 rad or 90º. Therefore,the commutative law of addition is not satisfied.