Lecture 004

Cross Product

Cross Product ( $\times$ ): perpendicular ( $\perp$ ) to all input vectors

obtained by solving system of equation of $\overrightarrow{w} \cdot x = 0 | \forall x$ input (only particular solution are cross product, because there might be potentially infinite many solutions)
$\overrightarrow{w} = \overrightarrow{u} \times \overrightarrow{v} = \det \begin{pmatrix} \overrightarrow{i} \overrightarrow{j} \overrightarrow{k}\\ u_1 u_2 u_3\\ v_1 v_2 v_3\\ \end{pmatrix}$
there are (at least) two vectors perpendicular to all input, let the first one be $x$ , second one be $y$ , by right hand rule, cross product must be in direction of $z$ , which means cross product is not commutative

Formal Determinant:

$\det \begin{pmatrix} \overrightarrow{i} & \overrightarrow{j} & \overrightarrow{k}\\ u_1 & u_2 & u_3\\ v_1 & v_2 & v_3\\ \end{pmatrix} = \overrightarrow{i} \det \begin{pmatrix} \overrightarrow{u_2} \overrightarrow{u_3}\\ \overrightarrow{v_2} \overrightarrow{v_3}\\ \end{pmatrix} - \overrightarrow{j} \det \begin{pmatrix} \overrightarrow{u_1} \overrightarrow{u_3}\\ \overrightarrow{v_1} \overrightarrow{v_3}\\ \end{pmatrix} + \overrightarrow{k} \det \begin{pmatrix} \overrightarrow{u_2} \overrightarrow{u_2}\\ \overrightarrow{v_2} \overrightarrow{v_2}\\ \end{pmatrix}$

Determinant: for 2 by 2, $\det \begin{pmatrix} a b\\ c d\\ \end{pmatrix} = ad - bc$

Properties of Cross Product:

Anticommutative: $\overrightarrow{u} \times \overrightarrow{v} = -(\overrightarrow{v} \times \overrightarrow{u})$
Distributive: $\overrightarrow{u} \times (\overrightarrow{v} + \overrightarrow{w}) = \overrightarrow{u} \times \overrightarrow{v} + \overrightarrow{u} \times \overrightarrow{w}$
Multiplication by Constant: $c(\overrightarrow{u} \times \overrightarrow{v}) = (c \overrightarrow{u}) \times \overrightarrow{v} = \overrightarrow{u} \times (c \overrightarrow{v})$
Zero Vector / Itself: $\overrightarrow{u} \times \overrightarrow{0} = \overrightarrow{0} \times \overrightarrow{u} = 0 = \overrightarrow{v} \times \overrightarrow{v}$
Scalar Triple Product: $\overrightarrow{u} \cdot (\overrightarrow{v} \times \overrightarrow{w}) = (\overrightarrow{u} \times \overrightarrow{v}) \cdot \overrightarrow{w}$

Angle: $\sin(\theta) = \frac{\|\overrightarrow{u} \times \overrightarrow{v}\|}{\|\overrightarrow{u}\|\|\overrightarrow{v}\|}$

Area of Parallelgram: $\|\overrightarrow{u} \times \overrightarrow{v}\|$ by applying the above formula

we can find volume of parallelepiped by triple scaler cross product

Torque: $\tau = \overrightarrow{r} \times \overrightarrow{F}$ where $\overrightarrow{r}$ is the radius and $\overrightarrow{F}$ is the force.

$\tau = \pm\|\overrightarrow{r}\| \|\overrightarrow{F}\| \sin{\theta}$ (positive is counterclockwise direction)

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