Vectors A and B are at angles α = 44.9° and β = 26.2° up from the x-axis respectively. If the vector sum A B C = 0, what are the magnitudes

Vectors A and B are at angles α = 44.9° and β = 26.2° up from the x-axis respectively. If the vector sum A B C = 0, what are the magnitudes of A and B?

a = magnitude of A,   b = magnitude of B

$$\small{
\vec{A} = a\cdot \begin{pmatrix}\cos{\alpha}\\\sin{\alpha}\end{pmatrix}\quad
\vec{B} = b\cdot \begin{pmatrix}\cos{\beta}\\\sin{\beta}\end{pmatrix}
\qquad
\vec{A}_{\bot }= \begin{pmatrix}-\sin{\alpha}\\\cos{\alpha}\end{pmatrix}\quad
\vec{B}_{\bot } =\begin{pmatrix}-\sin{\beta}\\\cos{\beta}\end{pmatrix}
}\\\\
\small{
\begin{array}{rcl}
a\vec{A}+b\vec{B}+\vec{c}&=&\vec{0}\quad | \quad \cdot \vec{B}_{\bot }\\
a\vec{A}\vec{B}_{\bot }+b\vec{B}\vec{B}_{\bot }+\vec{c}\vec{B}_{\bot }&=&\vec{0}\quad | \quad \vec{B}\cdot {\vec{B}_{\bot }=0 \\
a\vec{A}\vec{B}_{\bot }+\vec{c}\vec{B}_{\bot }&=&\vec{0} \\
a &=&\dfrac { -\vec{c}\vec{B}_{\bot } } {\vec{A}\vec{B}_{\bot }}\\
a &=& \vec{c}\cdot \begin{pmatrix}\frac{ \sin{\beta} }{ \sin{ (\alpha-\beta) }} \\\\ \frac{-\cos{\beta}}{\sin{ (\alpha-\beta) }} \end{pmatrix}
= \vec{c}\cdot \begin{pmatrix}1.37706788482\\ -2.79857147409\end{pmatrix}\\
\end{array}
}\\\\\\
\small{
\begin{array}{rcl}
a\vec{A}+b\vec{B}+\vec{c}&=&\vec{0}\quad | \quad \cdot \vec{A}_{\bot }\\
a\vec{A}\vec{A}_{\bot }+b\vec{B}\vec{A}_{\bot }+\vec{c}\vec{A}_{\bot }&=&\vec{0}\quad | \quad \vec{A}\cdot {\vec{A}_{\bot }=0 \\
b\vec{B}\vec{A}_{\bot }+\vec{c}\vec{A}_{\bot }&=&\vec{0} \\
b &=&\dfrac { -\vec{c}\vec{A}_{\bot } } {\vec{B}\vec{A}_{\bot }}\\
b &=& \vec{c}\cdot \begin{pmatrix}\frac{ -\sin{\alpha} }{ \sin{(\alpha-\beta)}} \\\\ \frac{\cos{\alpha}}{\sin{(\alpha-\beta)}} \end{pmatrix}
= \vec{c}\cdot \begin{pmatrix}-2.20163122333\\ 2.20932981047\end{pmatrix}\\
\end{array}
}$$