diff --git a/docs/mcv4u7.md b/docs/mcv4u7.md index 41a0e09..20f8bc7 100644 --- a/docs/mcv4u7.md +++ b/docs/mcv4u7.md @@ -527,6 +527,26 @@ To determine the **direction** of a cross product, the right-hand rule can be us - the index finger is the direction of the second vector - the palm faces the direction of the cross product +### Applications of dot and cross products + +A **triple scalar product** is the result of a cross product performed first then put in a dot product. +$$|\vec{c}\bullet(\vec{a}\times\vec{b})|$$ + +In a **parallelpiped**, or a three-dimensional shape with six faces each a parallelogram with an identical one opposite it, the volume is the triple scalar product of the distinct three vectors that make up its side lengths: +$$A=|\vec{c}\bullet(\vec{a}\times\vec{b})|$$ + +**Torque** ($\vec{\tau}$ or $\vec{M}$) is the ability to rotate an object — effectively angular/rotational force — and is the cross product of the **outward-pointing radius vector** ($\vec{r}$) and the **force** vector ($\vec{F}$). +$$ +\begin{align*} +\vec{\tau}&=\vec{r}\times\vec{F} \\ +&=|\vec{r}||\vec{F}|\sin\theta +\end{align*} +$$ + +(Source: Kognity) + +The direction of the torque can be found using the **right-hand rule**. + ## Resources - [IB Math Analysis and Approaches Syllabus](/resources/g11/ib-math-syllabus.pdf)