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Methods of Analysis I: Review of Vectors, Dyadics, Matrices, and Determinants
Published in Ronald L. Huston, Principles of Biomechanics, 2008
As the name implies, the scalar triple product is a product of three vectors resulting in a scalar. Let A, B, and C be vectors and as before, let ni (i = 1, 2, 3) be mutually perpendicular unit vectors so that A, B, and C may be expressed in the forms () A=ainiB=biniC=cini
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Published in W. M. Haynes, David R. Lide, Thomas J. Bruno, CRC Handbook of Chemistry and Physics, 2016
W. M. Haynes, David R. Lide, Thomas J. Bruno
where is the angle between V2 and V3 and is the angle between V1 and the normal to the plane of V2 and V3. The determinant indicates that it can be considered as the volume of the parallelepiped whose three determining edges are V1 , V2 , and V3. Note that cyclic permutation of the subscripts does not change the value of the scalar triple product: [V1 V2 V3 ] = [V2 V3 V1 ] = [V3 V1 V2 ] but [V1 V2 V3 ] = -[V2 V1 V3 ] and [V1 V1 V2 ] 0. The product V1 × (V2 × V3 ) defines the vector triple product. The parentheses are vital to the definition. V1 × (V2 × V3 ) = (V1 V3 )V2 - (V1 V2 )V3 = b2 b3 i a1 c2 c3 c2 c3 j b1 a2 a3 k c1
Vectors
Published in Jamal T. Manassah, Elementary Mathematical and Computational Tools For Electrical and Computer Engineers Using Matlab®, 2017
If the vectors’ u→, v→, and w→ original points are brought to the same origin, these three vectors define a parallelepiped. The absolute value of the scalar triple product can then be interpreted as the volume of this parallelepiped. We have shown earlier that v→×w→ is a vector that is perpendicular to both v→ and w→, and whose magnitude is the area of the base parallelogram. From the definition of the scalar product, dotting this vector with u→ will give a scalar that is the product of the area of the parallelepiped base multiplied by the parallelepiped height, whose magnitude is exactly the volume of the parallelepiped.
The linear combination of vectors implies the existence of the cross and dot products
Published in International Journal of Mathematical Education in Science and Technology, 2018
The following discussion was motivated by comments made by the reviewer, who noted that the scalar triple product is equal to zero when two of the vectors are equal, thus implying that the vector product of the two non-equal vectors is perpendicular to the other vector. This observation will be examined here. Given three vectors a, b and c, their scalar triple product is defined by a · (b × c). Writing the vector product in determinant form and then performing the dot product, we can write where the row elements are the corresponding vector components. One of the properties of determinants is that the interchange of any two of its rows changes its sign, not its value, which means that two interchanges do not affect the determinant. Therefore, (e.g. [12]). These results will be applied to two special cases. First, let c = a. Then so that a is perpendicular to a × b. Next, let c = b. Then so that b is also perpendicular to a × b. In both cases, we used the fact that, by definition, the cross product of a vector with itself is equal to the zero vector. These last two results, however, do not constitute independent proof of the perpendicularity of the cross product. Rather, they are direct consequences of Hamilton's definitions. For example, if the coordinate system is rotated such that a and b are in the plane of the unit vectors i′ and j′ in the rotated system, then, in that system (a′ × b′)∝k′. As a′ and b′ are of the form (c1i′ + c2j′) for some c1 and c2, we see that a′ · k′ = b′ · k′ = 0. Therefore, it follows that (a′ × b′) is perpendicular to a′ and b′. Finally, because the cross product was defined geometrically (see Equation (4)), its properties are not affected by coordinate rotations, and the perpendicularity relations just derived also apply to the vectors in the unrotated system.