Finding Parametric Equations

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

the line segment with endpoints (-1,3) and (3,-2)

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Identify the two endpoints of the line segment: \(A(-1, 3)\) and \(B(3, -2)\).

Recall that a parametric equation for a line segment between points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) can be written as: \(x(t) = x_1 + t(x_2 - x_1)\) \(y(t) = y_1 + t(y_2 - y_1)\), where the parameter \(t\) varies between 0 and 1.

Substitute the coordinates of points \(A\) and \(B\) into the parametric formulas: \(x(t) = -1 + t(3 - (-1))\) \(y(t) = 3 + t(-2 - 3)\).

Simplify the expressions inside the parentheses: \(x(t) = -1 + t(4)\) \(y(t) = 3 + t(-5)\).

Write the final parametric equations for the line segment as: \(x(t) = -1 + 4t\) \(y(t) = 3 - 5t\), with \(t\) in the interval \([0, 1]\) to trace the segment from \(A\) to \(B\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Parametric Equations of a Line Segment

Parametric equations express the coordinates of points on a line segment as functions of a parameter, usually t. For a segment between points A and B, the equations are x = x_A + t(x_B - x_A) and y = y_A + t(y_B - y_A), where t varies from 0 to 1 to trace the segment.

Vector Representation of Points

Points in the plane can be represented as vectors from the origin. The vector from point A to point B is found by subtracting their coordinates, which helps in defining direction and magnitude for parametrization of the line segment.

Parameter Interval and Its Role

The parameter t typically ranges between 0 and 1 to represent points along the line segment from the start to the end point. When t=0, the position corresponds to the first endpoint, and when t=1, it corresponds to the second endpoint.

Finding Parametric EquationsIn Exercises 31–36, find a parametrization for the curve.the line segment with endpoints (-1,3) and (3,-2)