Finding Parametric Equations

In Exercises 31–36, find a parametrization for the curve.

the ray (half line) with initial point (-1,2) that passes through the point (0,0)

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Identify the initial point of the ray as \(P_0 = (-1, 2)\) and the point it passes through as \(P_1 = (0, 0)\).

Find the direction vector \(\vec{d}\) of the ray by subtracting the coordinates of \(P_0\) from \(P_1\): \(\vec{d} = (0 - (-1), 0 - 2) = (1, -2)\).

Write the parametric equations for the ray starting at \(P_0\) and moving in the direction of \(\vec{d}\) using a parameter \(t \geq 0\): \(x(t) = -1 + 1 \cdot t\) and \(y(t) = 2 - 2 \cdot t\).

Specify the domain of the parameter \(t\) to ensure the curve is a ray (half-line), so \(t \geq 0\).

Thus, the parametric equations describe all points starting at \((-1, 2)\) and moving towards \((0, 0)\) and beyond in the direction of the vector \((1, -2)\).

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주요 개념

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Parametric Equations of a Line

Parametric equations express the coordinates of points on a line as functions of a parameter, usually t. For a line through a point P with direction vector d, the parametric form is x = x_0 + t * d_x and y = y_0 + t * d_y, where (x_0, y_0) is the initial point.

Direction Vector

The direction vector defines the line's orientation and is found by subtracting the initial point coordinates from another point on the line. For the ray starting at (-1, 2) and passing through (0, 0), the direction vector is (0 - (-1), 0 - 2) = (1, -2).

Parameter Domain for a Ray

A ray is a half-line starting at an initial point and extending infinitely in one direction. To represent a ray parametrically, the parameter t must be restricted to t ≥ 0, ensuring points lie on or beyond the initial point along the direction vector.

Finding Parametric EquationsIn Exercises 31–36, find a parametrization for the curve.the ray (half line) with initial point (-1,2) that passes through the point (0,0)