Dot Cross and Triple Products | Proof and Derivation

The dot product, cross product, and scalar triple product each serve a distinct purpose. The dot product provides a scalar value that measures the alignment of two vectors, while the cross product produces a new vector perpendicular to the original two. The scalar triple product, also a scalar, represents the volume of the parallelepiped formed by the three vectors.

The interactive demo enhances this understanding by transforming the static problem into a dynamic learning tool. By allowing users to change input values and instantly see the results, it bridges the gap between abstract, symbolic math and concrete, numerical outcomes. This real-time feedback loop helps to solidify the theoretical concepts and makes the learning process more intuitive and engaging.

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✍️Mathematical Proof

$\gg$Mathematical Structures Underlying Physical Laws

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Understanding Vector Components: The first step in solving any vector problem is to correctly identify the components of each vector. The problem's notation $v=x^2 e_1-x^1 e_2$ translates directly to the component form $v=\left(x^2,-x^1, 0\right)$, which is crucial for all subsequent calculations.
Dot Product as a Scalar: The dot product, $v \cdot w$, results in a scalar quantity ( $x^2 x^3$ in this case). It provides a measure of how much the two vectors point in the same direction. The calculation is a straightforward sum of the products of corresponding components.
Cross Product as a Vector: The cross product, $v \times w$, yields a new vector that is orthogonal (perpendicular) to both original vectors. The most systematic way to compute this is using the determinant of a $3 \times 3$ matrix, which correctly handles the direction and magnitude of the resulting vector. The result, $\left(x^1\right)^2 e_1+x^1 x^2 e_2+x^1 x^3 e_3$, is a vector field itself, with components that depend on the position ( $x^1, x^2, x^3$ ).
Scalar Triple Product as Volume: The scalar triple product, $v \cdot(w \times x)$, also known as the mixed product, results in a scalar quantity. Geometrically, its absolute value represents the volume of the parallelepiped defined by the three vectors. The calculation can be performed efficiently using the determinant of a $3 \times 3$ matrix formed by the vector components. The final simplified form, $x^1|x|^2$, shows that the volume of this parallelepiped is dependent on the $x^1$ component and the squared magnitude of the position vector.

🎬Demonstration

Interactive tools can transform a static, abstract problem into a dynamic and intuitive learning experience. Instead of just looking at a final, symbolic solution on paper, the interactive calculator allows you to:

Actively explore the problem by changing the input values.

Instantly see how those changes affect the outputs for the dot, cross, and scalar triple products.

Build a deeper understanding of vector operations by observing cause and effect in real-time, which helps to solidify the theoretical concepts.

The combination of a clean user interface and proper LaTeX rendering ensures that the focus remains on the math, making the tool a powerful educational aid for visualizing and verifying vector field calculations.

compute the dot product and cross product and scalar triple product