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Introduction to Linear Algebra
Published in Timothy Bower, ®, 2023
The cross product is a special operation using two vectors in ℝ3 with application to geometry and physical systems. Although cross product is an operation for vectors, it is included here because matrix operations (determinant or multiplication) is used to compute it.
Linear Algebra for Quantum Mechanics
Published in Caio Lima Firme, Quantum Mechanics, 2022
The cross product is an operation between two vectors which yields another vector in the direction of n, the unit vector perpendicular to the plane determined by vectors A and B. A×B=(‖A‖‖B‖sinθ)n
Fields that vary with time
Published in Andrew Norton, Dynamic Fields and Waves, 2019
The charges in the wire experience the magnetic part of the Lorentz force, which in a constant, uniform magnetic field may be expressed as: () F=q(v×B).What would be the direction of the force on a positive charge in the wire shown in Figure 1.6?To obtain the direction of a vector cross product, you need to use the right-hand rule. This is shown in Figure 1.7: you can see that a positive charge would experi-ence a force directed towards X.
Telemanipulation of cooperative robots: a case of study
Published in International Journal of Control, 2018
Javier Pliego-Jiménez, Marco Arteaga-Pérez
By taking into account (3) and (9) we have From (9) it is straightforward to show that On the other hand, according to Assumption 2.1 the vector can be computed as where and are orthogonal vectors as shown in Figure 2. By taking into account the previous equation, the total moment generated by the grasping force is given by Since the vector is parallel to , the cross product between these vectors is zero, From Equations (13) and (16) it follows From (13) and (17) it is clear that the grasping force does not contribute to the motion of the object. Therefore, the only forces and moments that contribute with its motion are given by Since is full-row rank matrix the solution of (18) for is given by where is the Moore-Penrose pseudoinverse. By taking into account (2), (10), (17)–(19) the dynamics of the object can be written as with and the following kinematic relationships have been used. By substituting Equation (20) into (6) we obtain the dynamics of the slave cooperative robotic system as where
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.