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Rakht CharitraQ:
Expressing the dot product of a vector and a 3 by 3 array of vectors
I am trying to express the dot product of a vector and a 3 by 3 array of vectors, however I am a bit stuck with how to do so. I have tried so far and have an answer, but it has been wrong so far. I am new to matrices and vectors in linear algebra, so any help would be greatly appreciated!
$$y=\begin{pmatrix}
1-\cos \alpha \\
\sin \alpha \\
\sin \alpha – \cos \alpha
\end{pmatrix}*\begin{pmatrix}
1 \\
\cos \beta \\
\sin \beta
\end{pmatrix}$$
Is the dot product of the first vector with the second and the third vector, i.e, $$\cos \alpha* 1+\sin \alpha* \cos \beta +\sin \alpha* \sin \beta $$
A:
Suppose
$$
\begin{pmatrix}
\cos\alpha & \sin\alpha & \sin\alpha-\cos\alpha
\end{pmatrix}
\begin{pmatrix}
1\\
\cos\beta\\
\sin\beta
\end{pmatrix}
$$
is a scalar multiple of a vector $y$; that is, $\cos\alpha*1+\sin\alpha*\cos\beta+\sin\alpha*\sin\beta = r$ for some scalar $r$. Then, since the dot product of a vector with a matrix is the trace of the matrix, we have
$$
\text{Tr}\begin{pmatrix}
\cos\alpha & \sin\alpha &
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