Finding the components of this vector

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The discussion centers on the transition from the vector representation of gravitational acceleration, denoted as g, to its scalar magnitude. Participants clarify that when referring to g, it represents the magnitude of the vector g, calculated as the square root of the sum of its components. Specifically, the scalar value of 9.8 m/s² is derived from the gravitational force acting in the vertical direction, simplifying the vector into a scalar for practical calculations. The confusion arises from not recognizing that g is the magnitude, while the vector form includes direction. Understanding this distinction is crucial for accurately applying gravitational concepts in physics.
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How is the vector g turning into the scalar g? What am I missing that is
allowing us to represent g as a scalar quantity and no longer a vector
quantity? When going through the algebra to find mg_x, (component
of gravity in the x-direction) I don't see how it turns into a
scalar of 9.8.

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When they say g instead of \vec{g} they mean the magnitude of \vec{g}, i.e. g=|\vec{g}| = \sqrt{g_x^2 + g_y^2}.
 

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