Final answer:
The parametric equation ⟨−5, 2⟩ + t⟨3, −1⟩ describes a line in the plane, where ⟨3, −1⟩ sets the line's direction and ⟨−5, 2⟩ is a point on the line.
Step-by-step explanation:
When we allow the scalar multiplying the vector ⟨3, −1⟩ to vary, the geometric object described by the parametric equation ⟨−5, 2⟩ + t⟨3, −1⟩ for all values of t is a line. This is because as the scalar t changes, the vector scales up and down in the direction of the vector ⟨3, −1⟩, which serves as the directional vector of the line. The point ⟨−5, 2⟩ provides the initial position of the vector, and hence, is a point on the line; varying t slides this point along the direction set by ⟨3, −1⟩. For positive values of t, the vector moves in the direction of ⟨3, −1⟩, and for negative values of t, the direction is reverse, which exhibits the concept of a vector being antiparallel when multiplied by a negative scalar.
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