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

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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.

✍️Mathematical Proof

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1. Proving the Cross Product Rules with the Levi-Civita Symbol
2. Proving the Epsilon-Delta Relation and the Bac-Cab Rule
3. Simplifying Levi-Civita and Kronecker Delta Identities
4. Dot Cross and Triple Products
5. Why a Cube's Diagonal Angle Never Changes
6. How the Cross Product Relates to the Sine of an Angle
7. Finding the Shortest Distance and Proving Orthogonality for Skew Lines
8. A Study of Helical Trajectories and Vector Dynamics
9. The Power of Cross Products: A Visual Guide to Precessing Vectors
10. Divergence and Curl Analysis of Vector Fields
11. Unpacking Vector Identities: How to Apply Divergence and Curl Rules
12. Commutativity and Anti-symmetry in Vector Calculus Identities
13. Double Curl Identity Proof using the epsilon-delta Relation
14. The Orthogonality of the Cross Product Proved by the Levi-Civita Symbol and Index Notation
15. Surface Parametrisation and the Verification of the Gradient-Normal Relationship
16. Proof and Implications of a Vector Operator Identity
17. Conditions for a Scalar Field Identity
18. Solution and Proof for a Vector Identity and Divergence Problem
19. Kinematics and Vector Calculus of a Rotating Rigid Body
20. Work Done by a Non-Conservative Force and Conservative Force
21. The Lorentz Force and the Principle of Zero Work Done by a Magnetic Field
22. Calculating the Area of a Half-Sphere Using Cylindrical Coordinates
23. Divergence Theorem Analysis of a Vector Field with Power-Law Components
24. Total Mass in a Cube vs. a Sphere
25. Momentum of a Divergence-Free Fluid in a Cubic Domain
26. Total Mass Flux Through Cylindrical Surfaces
27. Analysis of Forces and Torques on a Current Loop in a Uniform Magnetic Field
28. Computing the Integral of a Static Electromagnetic Field
29. Surface Integral to Volume Integral Conversion Using the Divergence Theorem
30. Circulation Integral vs. Surface Integral
31. Using Stokes' Theorem with a Constant Scalar Field
32. Verification of the Divergence Theorem for a Rotating Fluid Flow
33. Integral of a Curl-Free Vector Field
34. Boundary-Driven Cancellation in Vector Field Integrals
35. The Vanishing Curl Integral
36. Proving the Generalized Curl Theorem
37. Computing the Magnetic Field and its Curl from a Dipole Vector Potential
38. Proving Contravariant Vector Components Using the Dual Basis
39. Verification of Orthogonal Tangent Vector Bases in Cylindrical and Spherical Coordinates
40. Vector Field Analysis in Cylindrical Coordinates
41. Vector Field Singularities and Stokes' Theorem
42. Compute Parabolic coordinates-related properties
43. Analyze Flux and Laplacian of The Yukawa Potential
44. Verification of Vector Calculus Identities in Different Coordinate Systems
45. Analysis of a Divergence-Free Vector Field
46. The Uniqueness Theorem for Vector Fields
47. Analysis of Electric Dipole Force Field

🧄Proof and Derivation-2

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