Step 1 :The problem is asking for the x-value of the terminal point of a vector after it has been reflected using the given transformation matrix. The vector is not given, but we can assume it to be the vector [\(\sqrt{2}\), 0] since the x-value of the vector is given as \(\sqrt{2}\).
Step 2 :The transformation matrix given is a reflection matrix. This matrix reflects a vector across the x-axis. The x-values remain the same while the y-values are negated.
Step 3 :To find the x-value of the terminal point of the resulting vector, we need to multiply the vector by the transformation matrix. The resulting vector will have the x-value as the first element.
Step 4 :The x-value of the terminal point of the resulting vector after the transformation is the same as the x-value of the original vector, which is \(\sqrt{2}\). This is because the transformation matrix given is a reflection matrix that reflects a vector across the x-axis, and thus the x-values remain the same.
Step 5 :Final Answer: The x-value of the terminal point of the resulting vector is \(\boxed{\sqrt{2}}\).