Show that \(\mathbf{A} \times(\mathbf{B}+\mathbf{C})=\mathbf{A} \times \mathbf{B}+\mathbf{A} \times \mathbf{C}\)

When dealing with vectors, it's crucial to understand the rules that govern their interactions. One such rule is the distributive law of vector algebra, which dictates how vector multiplication is handled when multiple vectors are involved. Think of distribution as sharing or spreading something out. Mathematically, when we say that the cross product is distributive over addition, we mean that \( \mathbf{A} \times(\mathbf{B}+\mathbf{C}) \) can be split into \( \mathbf{A} \times \mathbf{B} + \mathbf{A} \times \mathbf{C} \).

This law is not only a theoretical concept but a practical tool when simplifying complex vector equations. To master its application, one should practice expanding cross products according to the distributive law and verifying results with known vector identities. This will help reinforce the understanding and fluency in working with vector operations, particularly when vectors are part of larger systems or equations.

The right-hand rule is a handy mnemonic for remembering the direction of the vector that results from a cross product. Imagine you're giving a 'thumbs-up' with your right hand. If you align your index finger with the first vector (\mathbf{A}) and your middle finger with the second vector (\mathbf{B}), your thumb points in the direction of the cross product \( \mathbf{A} \times \mathbf{B} \).

Practical Application

For students, using the right-hand rule often involves a physical gesture to visualize the result of a cross product. This visualization is crucial because it ties the abstract concept to a tangible action, making it easier to grasp and remember. Furthermore, the right-hand rule is not just a trick for students; it's used by professionals in fields like physics and engineering to quickly determine orientations in three-dimensional space.

Calculating the cross product of two vectors involves both direction and magnitude. The magnitude of the resulting vector from \( \mathbf{A} \times \mathbf{B} \) is given by the area of the parallelogram that they span. To calculate the components of the cross product, you would typically break down each vector into its i, j, k (unit vectors) components and use them in a determinant.

Step-by-Step Computation

When computing \( \mathbf{A} \times \mathbf{B} \), set up a 3x3 matrix with the unit vectors i, j, k in the first row, the components of \mathbf{A} in the second row, and the components of \mathbf{B} in the third row. The determinant of this matrix gives the components of the resulting cross product vector. Calculating cross products by hand in this way provides a solid foundation for understanding the geometric and algebraic properties of vectors, which can then be applied to more advanced problems in vector calculus and physical applications.