Question

a. Show that \(\mathbf{w} \cdot(\mathbf{u} \times \mathbf{v})=\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})=\mathbf{v} \times(\mathbf{w} \times \mathbf{u})\) holds for all vectors \(\mathbf{w}, \mathbf{u},\) and \(\mathbf{v}\) b. Show that \(\mathbf{v}-\mathbf{w}\) and \((\mathbf{u} \times \mathbf{v})+(\mathbf{v} \times \mathbf{w})+(\mathbf{w} \times \mathbf{u})\) are orthogonal.

Short answer

Both parts are verified using cyclic properties and orthogonality via dot product.

Step-by-step solution

Verifying the Vector Triple Product Identity

For part (a), we need to prove the vector identity \( \mathbf{w} \cdot (\mathbf{u} \times \mathbf{v}) = \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = \mathbf{v} \cdot (\mathbf{w} \times \mathbf{u}) \). According to the properties of the scalar triple product, these expressions are equal due to cyclic permutations of the dot and cross products. This is a known identity called the "scalar triple product identity," and it can be rearranged cyclically without changing its value.

Verify Orthogonality by Dot Product

For part (b), we need to demonstrate that \( \mathbf{v} - \mathbf{w} \) is orthogonal to \( (\mathbf{u} \times \mathbf{v}) + (\mathbf{v} \times \mathbf{w}) + (\mathbf{w} \times \mathbf{u}) \). Two vectors are orthogonal if their dot product is zero. Hence, we compute:\[(\mathbf{v} - \mathbf{w}) \cdot ((\mathbf{u} \times \mathbf{v}) + (\mathbf{v} \times \mathbf{w}) + (\mathbf{w} \times \mathbf{u}))\]Expanding the dot product, we have:\[(\mathbf{v} - \mathbf{w}) \cdot (\mathbf{u} \times \mathbf{v}) + (\mathbf{v} - \mathbf{w}) \cdot (\mathbf{v} \times \mathbf{w}) + (\mathbf{v} - \mathbf{w}) \cdot (\mathbf{w} \times \mathbf{u})\]Using the cyclic property of the scalar triple product,- \((\mathbf{v} - \mathbf{w}) \cdot (\mathbf{u} \times \mathbf{v}) = \mathbf{v} \cdot (\mathbf{u} \times \mathbf{v}) - \mathbf{w} \cdot (\mathbf{u} \times \mathbf{v}) = 0 \)- \((\mathbf{v} - \mathbf{w}) \cdot (\mathbf{v} \times \mathbf{w}) = 0 \)- \((\mathbf{v} - \mathbf{w}) \cdot (\mathbf{w} \times \mathbf{u}) = 0 \)Since all terms are zero, \( \mathbf{v} - \mathbf{w} \) is indeed orthogonal to \((\mathbf{u} \times \mathbf{v}) + (\mathbf{v} \times \mathbf{w}) + (\mathbf{w} \times \mathbf{u}) \).

Key concepts

Scalar Triple Product

The scalar triple product is a fascinating concept in vector calculus. It involves three vectors, say \( \mathbf{a}, \mathbf{b}, \mathbf{c} \), and is defined as \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \). This operation results in a scalar value, not another vector, hence the name 'scalar' triple product. The geometric interpretation of this product is the volume of the parallelepiped formed by the vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \). To appreciate how this works, think about the height being defined by the dot product and the base area expressed by the cross product.

One crucial property of the scalar triple product is its invariance under cyclical permutation. Thus, \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b}) \). This means the order does not matter in a cycle, resulting in the same value each time. Understanding this property helps simplify calculations and is widely used in proofs, such as the one in the original exercise.

Vector Cross Product

A vector cross product is a fundamental operation in vector calculus, producing a vector perpendicular to the plane formed by two initial vectors, like \( \mathbf{a} \times \mathbf{b} \). Given vectors \( \mathbf{a} \) and \( \mathbf{b} \), the magnitude of the resulting vector can be found by \( |\mathbf{a}| |\mathbf{b}| \sin(\theta) \), where \( \theta \) is the angle between \( \mathbf{a} \) and \( \mathbf{b} \). The direction of this vector follows the right-hand rule: curling the right fingers from \( \mathbf{a} \) to \( \mathbf{b} \) points the thumb in the direction of the cross product.

The cross product is antisymmetric, meaning \( \mathbf{a} \times \mathbf{b} = - (\mathbf{b} \times \mathbf{a}) \). Also notable is that the cross product is zero for parallel vectors because \( \sin(0^\circ) = 0 \), reflecting zero area for the parallelogram.
Applications of the cross product extend to physics and engineering to determine torque or rotational force, where magnitude and direction are crucial.

Orthogonality

Orthogonality is a concept indicating perpendicularity between vectors. In simpler terms, two vectors are orthogonal when their dot product equals zero, symbolizing no projection of one vector onto the other. For example, vectors \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal if \( \mathbf{a} \cdot \mathbf{b} = 0 \).

Let's unpack why this criteria works: the dot product formula \( |\mathbf{a}| |\mathbf{b}| \cos(\theta) \) indicates that if the vectors are perpendicular, then \( \theta = 90^\circ \), where \( \cos(90^\circ) = 0 \), thus making the product zero.
Understanding orthogonality is crucial in both theoretical and applied mathematics, especially in topics like projection, vector decomposition, and computer graphics. For instance, the orthogonal basis in linear algebra allows for simpler mapping of complex spaces. Overall, it plays an important part in many realms of science and engineering, facilitating calculations and interpretations of different physical systems.