PX153 - A6 - advanced vector operations

scalar triple product

defined by $[\vec{u}, \vec{v}, \vec{w}] \equiv \vec{u} \cdot (\vec{v} \times \vec{w})$
- the result is a scalar
- scalar triple product is the volume of the parallelepiped defined by $\vec{u}$ , $\vec{v}$ and $\vec{w}$
can be used to determine whether three vectors are linearly independent (lie in the same plane)
- if $[\vec{u}, \vec{v}, \vec{w}] = 0$ , they are linearly independent
$\vec{u} \cdot \vec{v} = \vec{v} \cdot \vec{u}$ but $\vec{u} \times \vec{v} = - \vec{c} \times \vec{u}$
- so, $\vec{u} (\vec{v} \times \vec{w}) = - (\vec{w} \times \vec{v}) \cdot \vec{u}$
to check: $\vec{u} (\vec{v} \times \vec{w}) = (\vec{u} \times \vec{v}) \cdot \vec{w}$
- in cartesian coordinates,
$R H S = (\vec{u} \times \vec{v}) \cdot \vec{w} = (u_{y} v_{z} - u_{z} v_{y}) w_{x} + (u_{z} v_{x} - u_{x} v_{z}) w_{y} + (u_{x} v_{y} - u_{y} v_{x}) w_{z}$
$= w_{x} u_{y} v_{z} - w_{x} v_{y} u_{z} + v_{x} w_{y} v_{z} - u_{x} w_{y} v_{z} + u_{x} v_{y} w_{z} - v_{x} u_{y} w_{z}$
$L H S = \vec{u} \cdot (\vec{v} \times \vec{w}) = (v_{y} w_{z} - v_{z} w_{y}) u_{x} + (v_{z} w_{x} - v_{x} w_{z}) u_{y} + (v_{x} w_{y} - v_{y} w_{x}) u_{z}$
$= w_{x} u_{y} v_{z} - w_{x} v_{y} u_{z} + v_{x} w_{y} v_{z} - u_{x} w_{y} v_{z} + u_{x} v_{y} w_{z} - v_{x} u_{y} w_{z}$
$∴ L H S = R H S$
alt method for computing:

[u, v, w] = det | \begin{matrix} u_{x} & u_{y} & u_{z} \\ v_{x} & v_{y} & v_{z} \\ w_{x} & w_{y} & w_{z} \end{matrix} |

to find: volume for the parallelepiped with vertices:
$A = (1, 3, 0)$
$B = (2, 1, 6)$
$C = (1, 1, 3)$
$D = (0, 1, 4)$
find $\vec{A B}, \vec{A C}, \vec{A D}$
volume = $[\vec{A D}, \vec{A C}, \vec{A B}]$
- order of vectors doesn't matter, just take the absolute value of the scalar triple product

\vec{u} \times (\vec{v} \times \vec{w}) \neq (\vec{u} \times \vec{v}) \times \vec{w})

\vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \cdot \vec{w}) \vec{v} - (\vec{u} . \vec{v}) \vec{w}

$\vec{v} \times \vec{w}$ gives a vector perpendicular to the plane $\vec{v}$ , $\vec{w}$
- if you work through the components, the above result can be proved.
- eg: if angular velocity $\vec{ω}$
  - centripetal acceleration: $\vec{a} = \vec{ω} \times (\vec{ω} \times \vec{r})$