How the Cross Product Relates to the Sine of an Angle

The relationship between the cross product and the geometry of two vectors reveals that the magnitude of their vector product is intrinsically linked to the area of the parallelogram they span. By expanding the squared magnitude of the cross product $|v \times w|^2$ into its Cartesian components, we can algebraically prove Lagrange's Identity, which demonstrates that this value is equivalent to the difference between the product of the squared magnitudes $|v|^2|w|^2$ and the square of the dot product $(v \cdot w)^2$. Consequently, by substituting the trigonometric identities for the dot and cross products into this relationship, we derive that the sine of the angle $\theta$ between the vectors is the ratio of the magnitude of the cross product to the product of the individual magnitudes. This provides a direct method to determine the angular orientation of vectors in 3D space using only their algebraic components.

🎬Narrated Video

https://youtu.be/UdwNcFBN_CI

🏗️Structural clarification of Poof and Derivation

block-beta
columns 5
CC["Criss-Cross"]:5

%% Condensed Notes

CN["Condensed Notes"]:5
RF["Relevant File"]:5
NV["Narrated Video"]:4 VO["Voice-over"] 
PA("Plotting & Analysis")AA("Animation & Analysis")KT("Summary & Interpretation") ID("Illustration & Demo") PO("Polyptych")

%% Proof and Derivation

PD["Proof and Derivation"]:5
AF("Derivation Sheet"):5
NV2["Narrated Video"]:4 VO2["Voice-over"]
PA2("Plotting & Analysis")AA2("Animation & Analysis")KT2("Summary & Interpretation") ID2("Illustration & Demo") PO2("Polyptych")

classDef color_1 fill:#8e562f,stroke:#8e562f,color:#fff
class CC color_1

%% %% Condensed Notes

classDef color_2 fill:#14626e,stroke:#14626e,color:#14626e
class CN color_2
class RF color_2

classDef color_3 fill:#1e81b0,stroke:#1e81b0,color:#1e81b0
class NV color_3
class PA color_3
class AA color_3
class KT color_3
class ID color_3

classDef color_4 fill:#47a291,stroke:#47a291,color:#47a291
class VO color_4
class PO color_4

%% Proof and Derivation

classDef color_5 fill:#307834,stroke:#307834,color:#fff
class PD color_5
class AF color_5

classDef color_6 fill:#38b01e,stroke:#38b01e,color:#fff
class NV2 color_6
class PA2 color_6
class AA2 color_6
class KT2 color_6
class ID2 color_6

classDef color_7 fill:#47a291,stroke:#47a291,color:#fff
class VO2 color_7
class PO2 color_7

🗒️Downloadable Files - Recursive updates

‣

Copyright Notice

All content and images on this page are the property of Sayako Dean, unless otherwise stated. They are protected by United States and international copyright laws. Any unauthorized use, reproduction, or distribution is strictly prohibited. For permission requests, please contact [email protected]

</aside>