# 3.14 Vector Analysis And Classical Identities in 3D

$$\def\pd{\partial}\def\Real{\mathbb{R}}$$

1. There are two main types of products between vectors in $$\mathbb{R}^3$$:
The inner/scalar/dot product
$$A\cdot B = A_x+B_x + A_yB_y + A_z Bz \in \mathbb{R}$$
is commutative, distributive, and homogenenous.
The vector (cross) product:
$$A\times B = \begin{pmatrix} A_yB_z-A_zB_y \\ A_zB_c-A_zB_z \\ A_xB_y-A_yB_x\end{pmatrix} \in \mathbb{R^3}$$
is homogeneous, not commutative, not associative, but linear with
respect to each entry.
2. The cross product $$A\times B$$ is always perpendicular to $$A$$ and $$B$$,
and its length equals to $$|A||B|\sin(\phi)$$ where $$\phi$$ is the angle
between the vectors. This leads to interesting geometrical uses. For example,
the product $$A\cdot (B\times C)$$ is the the volume of the parallelepiped
spanned by $$A,B$$ and $$C$$.
3. Another interesting identity:
$$A\times(B\times C) = (A\cdot C)B – (A\cdot B) C$$
(here $$\cdot$$ denotes the dot product).
4. The Jacobi identity the cross product is not associative, that
is $$(A\times B)\times C \neq A\times(B\times C)$$. However,
for every three vectors $$A,B,C\in \Real^3$$ we have the Jacobi identity:
$$A\times (B \times C) + B\times (C \times A) + C\times (A \times B) = 0$$
Recall that the three altitudes of a triangle intersect in a single point.
This result, incidentally, remains true for triangles bounded by big-circles
on a surface of a sphere. That is, the altitudes of a triangular patch on
a spherical shell intersect in a single point.
5. Exercise:
Assume that 4 vectors in $$\Real^2$$ satisfy $$A+B+C+D=0$$. Simplify
$A\times B-B\times C+C\times D-D\times A.$