Chapter 12: Vectors and the Geometry of Space

Chapter omitted because this is more of a linear algebra basics chapter. But, here's a nice reference sheet:

References

Dot Product

\begin{align*} u\cdot v&=\lvert u \rvert \lvert v \rvert \cos\theta \\ u\cdot v&=0\implies u\perp v \\ \\ u\cdot u&=\lvert u \rvert ^{2} \\ u\cdot v&=v\cdot u \\ u\cdot (v+w) &= u\cdot v+u\cdot w \\ (\alpha u)\cdot v&=\alpha(u\cdot v)=u\cdot (\alpha v) \\ \mathbf{0}\cdot u&=0 \\ \end{align*}

Projections

\begin{align*} \text{comp}_{u}v &= \frac{u\cdot v}{\lvert u \rvert } \\ \text{proj}_{u}v &= \left( \frac{u\cdot v}{\lvert u \rvert } \right) \frac{u}{\lvert u \rvert } \end{align*}

Cross Product

\begin{align*} (u\times v) &\perp u,v \\ \lvert u\times v \rvert &=\lvert u \rvert \lvert v \rvert \sin\theta \\ u\times v&=\mathbf{0} \implies u\perp v \\ \end{align*}

\begin{align*} \lvert u\times v \rvert &= \text{area of parallelogram determined by }u\text{ and }v \\ \end{align*}

\begin{align*} i\times j&=k &j\times k &=i &k\times i&=j \\ j\times i&=-k &k\times j&=-i &i\times k &=-j \end{align*}

\begin{align*} u\times v&=-u\times v \\