vector¶
Module: core
Category: mat3dvaluator
Type string: "vector"
Parameters¶
| Name | Description | Default | Units |
|---|---|---|---|
a |
a | {1,0,0} | [] |
d |
d | {0,1,0} | [] |
Description¶
The vector mat3d valuator generates a \(3 \times 3\) orthonormal matrix using the two vectors a and b defined by the user. It is assumed that a and d are not collinear.
First, let \(\hat{\mathbf{a}}=\mathbf{a}/\left||\mathbf{a} \right||\) and \(\hat{\mathbf{d}}=\mathbf{d}/\left||\mathbf{d} \right||\). Then, the three vectors are determined from,
\[\begin{align}
& {{\mathbf{e}}_{1}}=\hat{\mathbf{a}} \\
& {{\mathbf{e}}_{3}}=\hat{\mathbf{a}}\times \hat{\mathbf{d}} \\
& {{\mathbf{e}}_{2}}={{\mathbf{e}}_{3}}\times {{\mathbf{e}}_{1}} \\
\end{align}\]
From this, the orthonormal matrix is constructed,
\[
\mathbf{Q}=\left[ \begin{matrix} {\mathbf{e}_{1}} & {\mathbf{e}_{2}} & {\mathbf{e}_{3}} \end{matrix} \right]
\]