find their precursors and roots in vector algebra and calculus and hence many

Find their precursors and roots in vector algebra and

This preview shows page 24 - 27 out of 171 pages.

find their precursors and roots in vector algebra and calculus and hence many concepts, techniques and notations of the former can be viewed as extensions and generalizations of their counterparts in the latter. A. Dot Product of Vectors The dot product, or scalar product , of two vectors is a scalar quantity which has two interpretations: geometric and algebraic. Geometrically , the dot product of two vectors a and b can be interpreted as the projection of a onto b times the length of b , or as the projection of b onto a times the length of a , as demonstrated in Figure 6. In both cases, the dot product is obtained by taking the product of the length of the two vectors involved times the cosine of the angle between them when their tails are made to coincide, that is: a · b = | a | | b | cos θ (13) where the dot between a and b on the left hand side of the equation stands for the dot product operation, 0 θ π is the angle between the two vectors and the bars notation means the modulus or the length of the vector. Algebraically , the dot product is the sum of the products of the corresponding com- ponents of the two vectors, that is: a · b = a 1 b 1 + a 2 b 2 + a 3 b 3 (14) where a i and b j ( i, j = 1 , 2 , 3 ) are the components of a and b respectively. Here, we are assuming an orthonormal Cartesian system in a 3D space; the formula can be easily
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1.3.2 Vector Algebra and Calculus 24 θ | b | | a | cos θ θ | a | | b | cos θ θ a b Figure 6: Demonstration of the geometric interpretation of the dot product of two vectors a and b (left frame) as the projection of a onto b times the length of b (middle frame) or as the projection of b onto a times the length of a (right frame). extended to an n D space, that is: a · b = n X i =1 a i b i (15) From Eq. 13, it is obvious that the dot product is positive when 0 θ < π 2 , zero when θ = π 2 (i.e. the two vectors are orthogonal), and negative when π 2 < θ π . The magnitude of the dot product is equal to the product of the lengths of the two vectors when they have the same orientation (i.e. parallel or anti-parallel). Based on the above given facts, the dot product is commutative , that is: a · b = b · a (16) B. Cross Product of Vectors Geometrically , the cross product, or vector product , of two vectors, a and b , is a vector whose length is equal to the area of the parallelogram defined by the two vectors as its two main sides when their tails coincide and whose orientation is perpendicular to the plane of the parallelogram with a direction defined by the right hand rule as
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1.3.2 Vector Algebra and Calculus 25 demonstrated in Figure 7. Hence the cross product of two vectors a and b is given by: a × b = ( | a | | b | sin θ ) n (17) where 0 θ π is the angle between the two vectors when their tails coincide and n is a unit vector perpendicular to the plane containing a and b and is directed according to the right hand rule.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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