I'm not sure how the matrix multiplication at the second image works is it a typo or is my understanding of matrix multiplication flawed.Full solution
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$\begingroup$The cross product is defined as follows:
$$\begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}\times \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = \begin{bmatrix} b_2c_3-b_3c_2 \\ -(b_1c_3-b_3c_1) \\ b_1c_2-b_2c_1\end{bmatrix}$$
It take in two vectors and returns a vector that is perpendicular to the plane generated by the first two vectors.
The dot product is defined as follows:
$$\begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}\cdot \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = b_1c_1+b_2c_2+b_3c_3$$
Hence,
\begin{align}&\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\cdot \begin{bmatrix} \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} \times \begin{pmatrix} x_1+y_1 \\ x_2+y_2 \\ x_3+y_3 \end{pmatrix}\end{bmatrix}\\&= \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\cdot \begin{bmatrix} \begin{pmatrix} b_2(x_3+y_3)-b_3(x_2+y_2) \\ -b_1(x_3+y_3)+b_3(x_1+y_1) \\ b_1(x_2+y_2)-b_2(x_1+y_1) \end{pmatrix} \end{bmatrix} \\&=a_1 (b_2(x_3+y_3)-b_3(x_2+y_2))+a_2(-b_1(x_3+y_3)+b_3(x_1+y_1))+a_3(b_1(x_2+y_2)-b_2(x_1+y_1))\end{align}
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