The cross product of two vectors in ℝ^2 is defined as the scalar 𝐯 ×𝐰=v1 w2-v2 w1 for 𝐯=(v

step 1 Hello guys. So our question is use the definition of cross product to prove that the cross produ

The cross product of two vectors in ℝ^2 is defined as the...

The cross product of two vectors in $\mathbb{R}^2$ is defined as the scalar $$ \mathbf{v} \times \mathbf{w}=v_1 w_2-v_2 w_1 \quad \text { for } \quad \mathbf{v}=\left(v_1, v_2\right)^T, \quad \mathbf{w}=\left(w_1, w_2\right)^T . $$ (a) Does the cross product define an inner product on $\mathbb{R}^2$ ? Carefully explain which axioms are valid and which are not. (b) Prove that $\mathbf{v} \times \mathbf{w}=\|\mathbf{v}\|\|\mathbf{w}\| \sin \theta$, where $\theta$ denotes the angle from $\mathbf{v}$ to $\mathbf{w}$ as in Figure 3.2. (c) Prove that $\mathbf{v} \times \mathbf{w}=0$ if and only if $\mathbf{v}$ and $\mathbf{w}$ are parallel vectors. (d) Show that $|\mathbf{v} \times \mathbf{w}|$ equals the area of the parallelogram defined by $\mathbf{v}$ and $\mathbf{w}$.

The cross product of two vectors in $\mathbb{R}^2$ is defined as the scalar
$$
\mathbf{v} \times \mathbf{w}=v_1 w_2-v_2 w_1 \quad \text { for } \quad \mathbf{v}=\left(v_1, v_2\right)^T, \quad \mathbf{w}=\left(w_1, w_2\right)^T .
$$
(a) Does the cross product define an inner product on $\mathbb{R}^2$ ? Carefully explain which axioms are valid and which are not. (b) Prove that $\mathbf{v} \times \mathbf{w}=\|\mathbf{v}\|\|\mathbf{w}\| \sin \theta$, where $\theta$ denotes the angle from $\mathbf{v}$ to $\mathbf{w}$ as in Figure 3.2. (c) Prove that $\mathbf{v} \times \mathbf{w}=0$ if and only if $\mathbf{v}$ and $\mathbf{w}$ are parallel vectors. (d) Show that $|\mathbf{v} \times \mathbf{w}|$ equals the area of the parallelogram defined by $\mathbf{v}$ and $\mathbf{w}$.

Key Concepts

Inner Product Axioms

Inner product spaces are defined by axioms such as linearity in each argument (or at least one of them), symmetry (or conjugate symmetry in the complex case), and positive definiteness, meaning that the inner product of a vector with itself is always nonnegative and equals zero only for the zero vector. These conditions are essential for establishing a notion of length and angle in a vector space, which is why checking these properties is key when determining whether a given operation qualifies as an inner product.

Determinant and its Application in 2D Vector Operations

In two-dimensional space, the cross product can be defined using the determinant of a 2×2 matrix constructed from two vectors. This determinant captures both the signed area and orientation of the parallelogram defined by the vectors. The use of the determinant provides a systematic way to measure this signed area in a manner that reflects more than just size—it also conveys directional (or orientation) information.

Relationship Between Sine of the Angle and the Cross Product

A fundamental geometric interpretation of the cross product is that its magnitude equals the product of the magnitudes of the two vectors and the sine of the angle between them. This shows that the cross product is not only an algebraic tool but also provides insight into the angular relationship of vectors. Such a relationship is fundamental in relating algebraic operations in vector spaces to their geometric properties.

Area of a Parallelogram via the Cross Product

The magnitude of the cross product of two vectors corresponds exactly to the area of the parallelogram formed by those vectors. This is because the product of the vectors’ lengths multiplied by the sine of the angle between them gives the height of the parallelogram corresponding to the base represented by one of the vectors, thereby connecting the algebraic definition of the cross product directly to a key geometric measurement.

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