# How the relationship between the cross product and the geometry of two vectors applies in physics to

> The animated demonstrations for Magnetic Force ($$\vec{F}\_B=q(\vec{v} \times \vec{B})$$) and Torque ($$\vec{\tau}=\vec{r} \times \vec{F}$$) both beautifully illustrate the critical role of the cross product in physics, specifically confirming the trigonometric relationship $$|\vec{v} \times \vec{w}|=|\vec{v}||\vec{w}| \sin \theta$$. In both scenarios, the magnitude of the resultant vector-the magnetic force in the first case and the rotational twisting force (torque) in the second-was shown to be directly proportional to the sine of the angle ($$\theta$$) between the two input vectors. This means the effect is maximal when the input vectors (e.g., velocity and field, or lever arm and force) are perpendicular ( $$\sin 90^{\circ}=1$$ ), and the effect is zero when the input vectors are parallel or antiparallel ( $$\sin 0^{\circ}=0$$ ), thus providing a clear visual and algebraic confirmation that the cross product precisely models phenomena that rely on perpendicular interaction.

### :clapper:Narrated Video

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### :thread:Related Derivation

{% content-ref url="../proof-and-derivation/how-the-cross-product-relates-to-the-sine-of-an-angle-cp-sa" %}
[how-the-cross-product-relates-to-the-sine-of-an-angle-cp-sa](https://via-dean.gitbook.io/all/~/revisions/WYsapCOkHSgcDPhymfq8/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/proof-and-derivation/how-the-cross-product-relates-to-the-sine-of-an-angle-cp-sa)
{% endcontent-ref %}

### :hammer\_pick:Compound Page

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