From Eq 17 it can be seen that the cross product vector is zero when θ 0or θ π

From eq 17 it can be seen that the cross product

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From Eq. 17, it can be seen that the cross product vector is zero when θ = 0 or θ = π , i.e. when the two vectors are parallel or anti-parallel respectively. Also, the length of the cross product vector is equal to the product of the lengths of the two vectors when the two vectors are orthogonal (i.e. θ = π 2 ). b θ a a × b | a × b | = | a | | b | sin θ a b a × b Figure 7: Graphical demonstration of the cross product of two vectors a and b (left frame) with the right hand rule (right frame). Algebraically , the cross product of two vectors a and b is expressed by the following determinant (see § Determinant of Matrix) where the determinant is expanded along its
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1.3.2 Vector Algebra and Calculus 26 first row, that is: a × b = e 1 e 2 e 3 a 1 a 2 a 3 b 1 b 2 b 3 = ( a 2 b 3 - a 3 b 2 ) e 1 + ( a 3 b 1 - a 1 b 3 ) e 2 + ( a 1 b 2 - a 2 b 1 ) e 3 (18) Here, we are assuming an orthonormal Cartesian coordinate system in a 3D space with a basis vector set e 1 , e 2 and e 3 . Based on the above given facts, since the direction is determined by the right hand rule the cross product is anti-commutative , that is: a × b = - b × a (19) C. Scalar Triple Product of Vectors The scalar triple product of three vectors ( a , b and c ) is a scalar quantity defined by the expression: a · ( b × c ) (20) where the dot and multiplication symbols stand respectively for the dot and cross product operations of two vectors as defined above. Hence, the scalar triple product is defined geometrically by: a · ( b × c ) = | a | | b | | c | sin φ cos θ (21) where φ is the angle between b and c while θ is the angle between a and b × c . The scalar triple product is illustrated graphically in Figure 8. As there is no meaning of a cross product operation between a scalar and a vector, the parentheses in the above equation are redundant although they provide a clearer and more clean notation. Now, since | b × c | (= | b | | c | sin φ ) is equal to the area of the parallelogram whose two
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1.3.2 Vector Algebra and Calculus 27 a b c b × c φ θ Figure 8: Graphic illustration of the scalar triple product of three vectors a , b and c . The magnitude of this product is equal to the volume of the seen parallelepiped. main sides are b and c , while | a | cos θ represents the projection of a onto the orientation of b × c and hence it is equal to the height of the parallelepiped (refer to Figure 8), the magnitude of the scalar triple product is equal to the volume of the parallelepiped whose three main sides are a , b and c while its sign is positive or negative depending, respectively, on whether the vectors a , b and c form a right-handed or left-handed system. The scalar triple product is invariant to a cyclic permutation of the symbols of the three vectors involved, that is: a · ( b × c ) = c · ( a × b ) = b · ( c × a ) (22) It is also invariant to an exchange of the dot and cross product symbols, that is: a · ( b × c ) = ( a × b ) · c (23)
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1.3.2 Vector Algebra and Calculus 28 Hence, from the three possibilities of the first invariance with the two possibilities of the second invariance, we have six equal expressions for the scalar triple product of three vectors. [10] The other six possibilities of the scalar triple product of three vectors, which
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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