Final Answer:
(a) Parallel to the graph of f(x) at the point (2, 3) is the unit vector ⟨2, 1⟩, since the graph of f(x) passes through this point and has a slope of 1.
(b) Orthogonal to the graph of f(x) at the point (2, 3) is the unit vector ⟨1, 2⟩, since this vector is perpendicular to the tangent line of the graph at that point.
Step-by-step explanation:
To find a unit vector parallel to the graph of a function f(x) at a point (a, b), we need to find a vector that has the same direction as the tangent line of the graph at that point. In other words, we need to find a vector that has the same slope as the function at that point.
At the point (2, 3), the function f(x) has a slope of 1, so a unit vector parallel to the graph of f(x) at that point would be one with a slope of 1. In other words, a unit vector parallel to the graph of f(x) at (2, 3) is ⟨2, 1⟩.
To find a unit vector orthogonal to the graph of a function f(x) at a point (a, b), we need to find a vector that is perpendicular to the tangent line of the graph at that point. In other words, we need to find a vector that has a slope that is opposite to the slope of the function at that point.
At the point (2, 3), the function f(x) has a slope of 1, so a unit vector orthogonal to the graph of f(x) at that point would be one with a slope of -1. In other words, a unit vector orthogonal to the graph of f(x) at (2, 3) is ⟨1, 2⟩.
These two unit vectors, ⟨2, 1⟩ and ⟨1, 2⟩, are the desired parallel and orthogonal unit vectors, respectively, at the point (2, 3) for the function f(x).