multyplication of 2 vectors forming a matrix - meaning

Question

I am trying understand an algorithm used to determine orientations. Knowing a cross product of 2 vectors gives you a third vector which is orthogonal. What does the multiplication of a 3x1 and 1x3 matrix/vector represent? This should form a 3x3 matrix.

example equation 4.66 in the following context

(Vkb and Vki are vectors)

Answer 1

Yes, you're right with that. Note that for matrix multiplication you always have

$\underbrace{A}_{\in\mathbb{K}^{m\times n}} \cdot \underbrace{B}_{\in\mathbb{K}^{n\times k}} = C\in\mathbb{K}^{m\times k}$.

Your vector-vector multiplication (the vector is in $\mathbb{K}^{3\times 1}$) in the form $vv^T$ where $v=\begin{bmatrix}v_1\\v_2\\v_3\end{bmatrix}$ gives you indeed a matrix:

$\begin{bmatrix}v_1\\v_2\\v_3\end{bmatrix}\cdot\begin{bmatrix}v_1&v_2&v_3\end{bmatrix} = \begin{bmatrix}v_1v_1&v_1v_2&v_1v_3\\v_2v_1&v_2v_2&v_2v_3\\v_3v_1&v_3v_1&v_3v_3\end{bmatrix}$

EDIT: You were asking for a meaning of this, but I've never heard that this construction has a special meaning.

Answer 2

The matrix formed by the product $\mathbf v\mathbf w^T$ represents a linear transformation that maps the entire space onto the line containing the vector $\mathbf v$. If $\|\mathbf v\|=1$, then $\mathbf v\mathbf v^T$ represents orthogonal projection onto $\mathbf v$ (if $\|\mathbf v\|\ne1$, you can of course get orthogonal projection by dividing by $\|\mathbf v\|^2$).