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
    \[
    A\times B-B\times C+C\times D-D\times A.
    \]
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