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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
Vector algebra II Scalar and vector products
Published in A.V. Durrant, Vectors in Physics and Engineering, 2019
The vector product a × b is itself a vector. We can therefore form a scalar product of the vector a × b with a third vector c to give the scalar quantity (a × b). c. Actually the brackets can be removed without ambiguity, (a × b). c = a × b . c, because b . c is a scalar and so a × (b . c) is meaningless. This way of forming a product of three vectors to give a scalar is called a scalar triple product,
Force Analysis on Linkages
Published in Eric Constans, Karl B. Dyer, Introduction to Mechanism Design, 2018
If the preceding seems a little “mathemagical” to you, here is an alternative approach. The scalar triple product is often used in mathematics to calculate the volume of a parallelpiped (a solid body of which each face is parallelogram).
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.