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

How the relationship between the cross product and the geometry of two vectors applies in physics to calculate torque or magnetic force

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.