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Section 13.3 The dot product and cross product
Objectives
Use dot and cross products to capture alignment, perpendicularity, and orientation.
Compute these products from components to analyze motion and geometry in two and three dimensions.
Use normal vectors derived from cross products to express equations of planes.
In 2D, if
\(\bv{u}=(a,b)\) and
\(\bv{v}=(c,d)\) with angle
\(\theta\) between them (from
\(\bv{v}\) to
\(\bv{w}\text{,}\) then we define the dot product and cross product as follows:
\begin{align*}
\bv{u}\cdot\bv{v} \amp = \norm\big{u}\norm\big{v}\cos\theta\\
\bv{u}\times\bv{v} \amp = \norm\big{u}\norm\big{v}\sin\theta
\end{align*}
Hence the dot product is maximized when the vectors are parallel, and the cross product is maximized when the vectors are perpendicular.
Calculated componentwise this gives us:
\begin{align*}
\bv{u}\cdot\bv{v} \amp = ac+bd\\
\bv{u}\times\bv{v} \amp = ad-bc
\end{align*}
Show that an object in circular motion has velocity perpendicular to position and acceleration parallel to position.
In 3D, if
\(\bv{u}=(a,b,c)\) and
\(\bv{v}=(d,e,f)\) with angle
\(\theta\) between them (from
\(\bv{u}\) to
\(\bv{v}\text{,}\) then we define:
\begin{align*}
\bv{u}\cdot\bv{v} \amp = \norm\big{u}\norm\big{v}\cos\theta\\
\bv{u}\times\bv{v} \amp = (\norm\big{u}\norm\big{v}\sin\theta)\hat{\bv{w}}
\end{align*}
where
\(\hat{\bv{w}}\) is the unit vector that completes a right-handed coordinate system.
\begin{align*}
\bv{u}\cdot\bv{v} \amp = ad+be+cf\\
\bv{u}\times\bv{v} \amp = (bf-ce,cd-af,ae-bd)
\end{align*}
If
\(\DS\dv{t}(\bv{r}\times\bv{r}')=0\text{,}\) then the motion stays in a plane.
Need to include the relationship between planes and their normals here. Uses dot products, and the cross product can be used to find the normal to two given vectors.
