Question

Show that \((\mathbf{u} \times \mathbf{v}) \cdot(\mathbf{w} \times \mathbf{z})=\operatorname{det}\left[\begin{array}{cc}\mathbf{u} \cdot \mathbf{w} & \mathbf{u} \cdot \mathbf{z} \\ \mathbf{v} \cdot \mathbf{w} & \mathbf{v} \cdot \mathbf{z}\end{array}\right]\).

Short answer

The expression equals the determinant: \((\mathbf{u} \cdot \mathbf{w})(\mathbf{v} \cdot \mathbf{z}) - (\mathbf{u} \cdot \mathbf{z})(\mathbf{v} \cdot \mathbf{w})\)."

Step-by-step solution

Understand the Problem

We need to show that the dot product of two cross products \((\mathbf{u} \times \mathbf{v})\) and \((\mathbf{w} \times \mathbf{z})\) equals the determinant of a 2x2 matrix formed with dot products of the given vectors.

Recall the Vector Triple Product Identity

The result can be proved using the vector triple product identity: \((\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d}) = (\mathbf{a} \cdot \mathbf{c})(\mathbf{b} \cdot \mathbf{d}) - (\mathbf{a} \cdot \mathbf{d})(\mathbf{b} \cdot \mathbf{c})\). This identity gives us an expression for the dot product of two cross products.

Apply the Identity to the Given Expression

Let's apply this identity to our problem:\((\mathbf{u} \times \mathbf{v}) \cdot (\mathbf{w} \times \mathbf{z})\),where \(\mathbf{a} = \mathbf{u}, \mathbf{b} = \mathbf{v}, \mathbf{c} = \mathbf{w}, \mathbf{d} = \mathbf{z}\). Using the identity, we get:\((\mathbf{u} \cdot \mathbf{w})(\mathbf{v} \cdot \mathbf{z}) - (\mathbf{u} \cdot \mathbf{z})(\mathbf{v} \cdot \mathbf{w})\).

Write it as a Determinant

Recognize that the expression is equivalent to a determinant of a 2x2 matrix:\[\operatorname{det}\left[\begin{array}{cc} \mathbf{u} \cdot \mathbf{w} & \mathbf{u} \cdot \mathbf{z} \ \mathbf{v} \cdot \mathbf{w} & \mathbf{v} \cdot \mathbf{z} \end{array}\right] = (\mathbf{u} \cdot \mathbf{w})(\mathbf{v} \cdot \mathbf{z}) - (\mathbf{u} \cdot \mathbf{z})(\mathbf{v} \cdot \mathbf{w})\].Thus, we have shown that the equality holds.

Key concepts

Determinant

The concept of a determinant can be a bit abstract, but it is an essential part of many mathematical procedures, including the one used in this exercise. A determinant is a special number that can be calculated from a square matrix and provides important properties of the matrix, such as indicating whether the matrix is invertible. In simple terms, for a 2x2 matrix:\[\begin{bmatrix}a & b \c & d\end{bmatrix}\]The determinant is calculated as \(ad - bc\).
In the context of the exercise, we have a 2x2 matrix with elements being the dot products of vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w},\) and \(\mathbf{z}\). Calculated as:\[\operatorname{det}\left[\begin{array}{cc}\mathbf{u} \cdot \mathbf{w} & \mathbf{u} \cdot \mathbf{z} \mathbf{v} \cdot \mathbf{w} & \mathbf{v} \cdot \mathbf{z}\end{array}\right] = (\mathbf{u} \cdot \mathbf{w})(\mathbf{v} \cdot \mathbf{z}) - (\mathbf{u} \cdot \mathbf{z})(\mathbf{v} \cdot \mathbf{w})\]This result is the determinant, which shows how the values of these dot products are combined. The determinant reflects how linearly independent or dependent the two row vectors are. If the determinant is zero, it indicates linear dependence.

Dot Product

The dot product is a fundamental operation in vector algebra that produces a scalar result from two vectors. It's a measure of how much two vectors point in the same direction. Calculating the dot product involves multiplying corresponding components of two vectors and then adding those products together.
For example, for two vectors \(\mathbf{a} = (a_1, a_2, a_3)\) and \(\mathbf{b} = (b_1, b_2, b_3)\), the dot product \(\mathbf{a} \cdot \mathbf{b}\) is computed as:\[a_1b_1 + a_2b_2 + a_3b_3\]In the setting of the problem, dot products like \(\mathbf{u} \cdot \mathbf{w}\) and \(\mathbf{v} \cdot \mathbf{z}\) serve to "flatten" the multidimensional structure of vectors into simpler numerical expressions. This simplification allows us to easily place these results into a matrix, such as when determining if two cross products are equivalent when expressed as a determinant.The properties of the dot product, such as distributivity and commutativity, are critical when manipulating these operations to form solutions in vector calculus.

Cross Product

The cross product is another key operation in vector arithmetic. It takes two vectors and returns a third vector that is perpendicular to the plane formed by the first two. Unlike the dot product which results in a scalar, the cross product results in a vector. This is particularly useful in physics and engineering to determine angles and force directions.
The cross product for vectors \(\mathbf{a} = (a_1, a_2, a_3)\) and \(\mathbf{b} = (b_1, b_2, b_3)\) is determined as:\[\mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)\]When addressing the vector triple product identity stated in the exercise, the cross product acts as an essential component, manipulating the vectors into orientations that facilitate easier calculations, such as when using the identity to relate cross and dot products.
In essence, the cross product transforms spatial relationships into concrete results, assisting in proving more complex vector identities and understandings, such as the determinant representation in the given problem.