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Calculating the Cross Product of Two Vectors

Calculating the Cross Product of Two Vectors
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2016-05-18 15:45
VectorSolid GeometryGeometric Construction
This example demonstrates the calculation and geometric interpretation of the cross product of two vectors in three-dimensional space. Key math concepts include vectors, 3D space, cross product, vector magnitude and direction. It utilizes GeoGebra commands and objects such as Intersect, Vector, TableText, Plane, Angle, Segment, and LaTeX. Suitable for high school and above students learning vector operations and solid geometry.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, zAxis) → A
  2. Vector(A, B) → u
  3. Vector(A, C) → v
  4. TableText({{LaTeX["\hat{i}"], LaTeX["\hat{j}"], LaTeX["\hat{k}"]}, {LaTeX[(("\textcolor{#000099}{" * x(B)) * "}")], LaTeX[(("\textcolor{#000099}{" * y(B)) * "}")], LaTeX[(("\textcolor{#000099}{" * z(B)) * "}")]}, {LaTeX[(("\textcolor{#CC0000}{" * x(C)) * "}")], LaTeX[(("\textcolor{#CC0000}{" * y(C)) * "}")], LaTeX[(("\textcolor{#CC0000}{" * z(C)) * "}")]}}, "||c") → text2
  5. Plane(A, B, C) → a
  6. Angle(u, crossProduct) → α
  7. Angle(v, crossProduct) → β
  8. Segment((x(crossProduct), y(crossProduct), z(crossProduct)), D) → b
  9. Segment(D, F) → c
  10. Segment(D, E) → d
  11. Segment(B, G) → e
  12. Segment(H, G) → f
  13. Segment(I, G) → g
  14. Segment(C, J) → h
  15. Segment(K, J) → i
  16. Segment(L, J) → j
  17. LaTeX((((((((((((((((((((((((((((((((("\textcolor{#000099}{\vec{u}}\times\textcolor{#CC0000}{\vec{v}}\cr=\text{ }" * text2) * " \cr=\hat{i}[(\textcolor{#000099}{") * y(B)) * "})(\textcolor{#CC0000}{") * z(C)) * "})-(\textcolor{#000099}{") * z(B)) * "})(\textcolor{#CC0000}{") * y(C)) * "})] \cr\text{ }-\hat{j}[(\textcolor{#000099}{") * x(B)) * "})(\textcolor{#CC0000}{") * z(C)) * "})-(\textcolor{#000099}{") * z(B)) * "})(\textcolor{#CC0000}{") * x(C)) * "})] \cr\text{ }+\hat{k}[(\textcolor{#000099}{") * y(B)) * "})(\textcolor{#CC0000}{") * z(C)) * "})-(\textcolor{#000099}{") * z(B)) * "})(\textcolor{#CC0000}{") * y(C)) * "})] \cr=(") * x(crossProduct)) * ",") * y(crossProduct)) * ",") * z(crossProduct)) * ")")) → text3