12.4 Cross Product

Determinants

A determinant of order 2 is defined by $\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc$ A determinant of order 3 is defined by $\begin{vmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{vmatrix}=a_{1}\begin{vmatrix}b_{2}&b_{3}\\c_{2}&c_{3}\end{vmatrix}-a_{2}\begin{vmatrix}b_{1}&b_{3}\\c_{1}&c_{3}\end{vmatrix}+a_{3}\begin{vmatrix}b_{1}&b_{2}\\c_{1}&c_{2}\end{vmatrix}$

Cross Product

The cross product of $\vec{a}$ and $\vec{b}$ is a vector, sometimes called the "vector product". The product is a vector that is perpendicular to $\vec{a}$ and $\vec{b}$. It is calculated with a determinant 3x3 matrix that includes the standard basis vector and the components of the two vectors. $\vec{a}\times\vec{b}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\end{vmatrix}$

• To visualize which direction the cross product is in, use the right hand rule:
• $\vec{a}\times\vec{b}$ is orthogonal to $\vec{a}$ and $\vec{b}$.
• If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ ($0\leq\theta\leq\pi$), then the magnitude of the cross product, $|\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|\sin{\theta}$
  • Corollary: nonzero vector $\vec{a}$ and $\vec{b}$ are parallel if $|\vec{a}\times\vec{b}|=0$ because the angle between the vectors is $0^{\circ}$ or $180^{\circ}$.

Area of Traingle

Calculate the area of a trangle with points $P(1,4,6)$, $Q(-2,5,1)$, and $R(1,-1,1)$. $\vec{PQ}=<-2-1,5-4,1-6>=<-3,1,7>$ and $\vec{PR}=<0,-5,-5>$ $A=\frac{1}{2}bh=\frac{1}{2}ab\sin{\theta}$ $A=\frac{1}{2}|\vec{PQ}||\vec{PR}|\sin{P}$ $A=\frac{1}{2}|\vec{PQ}\times\vec{PR}|$ $A=\frac{1}{2}\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\-3&1&-7\\0&-5&5\end{vmatrix}$ $A=\frac{1}{2}|<-40,-15,15>|=\frac{1}{2}\sqrt{(-40)^2+(-15)^2+(15^2)}$ $A=\frac{5}{2}\sqrt{82}$

Properties of Cross Product

If $\vec{a}$, $\vec{b}$, and $\vec{c}$ are vectors and $c$ is a scalar:

1. $\vec{a}\times\vec{b}$=$-\vec{b} \times \vec{a}$
2. $c\vec{a}\times\vec{b}=c(\vec{a}\times\vec{b})=\vec{a}\times c\vec{b}$
3. $\vec{a}\times (\vec{b}+\vec{c}) = \vec{a}\times\vec{b}+\vec{a}\times \vec{c}$
4. $(\vec{a}+\vec{b}) \times c = \vec{a} \times c + \vec{b} \times c$
5. $\vec{a}\cdot(\vec{b} \times \vec{c}) = ({\vec{a}\times\vec{b}}) \cdot \vec{c}$
6. $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$
7. Cross product is not commutative.

Triple Products

$\vec{a} \cdot (\vec{b} \times \vec{c})$ is called the "scalar triple product" $\vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{vmatrix} \implies \mathbb{R}$

The volume of a parallelpiped determined by the vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ is the magnitude (always positive like volume) of their scalar triple product. $V= |\vec{a} \cdot (\vec{b} \times \vec{c})|$ If the volume is zero, that means that the vectors are coplanar, or on the same plane.

Torque

The torque vector ($\vec{\tau}$) is the coss product of position ($\vec{r}$) and force ($\vec{F}$) vector. $\vec{\tau} = \vec{r} \times \vec{F}$ Torque measures the tendency of the body to rotate about the origin. The direction of $\vec{\tau}$ is the axis of rotation. $|\vec{\tau}|=|\vec{r} \times \vec{F}| = |\vec{r}||\vec{F}|\sin{\theta}$ The magnitude of torque is equal to the area of the parallelogram determined by $\vec{r}$ and $\vec{F}$.