Question

find a parametric vector equation for the segment joining: $\mathbf{R}(2,5)$ and the midpoint of the segment with endpoints $\mathbf{S}(5,1)$ and $\mathbf{T}(7,-3)$.

   find a parametric vector equation for the segment joining:
$\mathbf{R}(2,5)$ and the midpoint of the segment with endpoints $\mathbf{S}(5,1)$ and $\mathbf{T}(7,-3)$.

Modern Analytic Geometry

Modern Analytic Geometry

William Wooton,… 1st Edition

Chapter 2, Problem 19

Instant Answer

Step 1

The formula for the midpoint \(\mathbf{M}\) of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ \mathbf{M} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substituting the coordinates of \(\mathbf{S}\) and \(\mathbf{T}\): Show more…

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find a parametric vector equation for the segment joining: $\mathbf{R}(2,5)$ and the midpoint of the segment with endpoints $\mathbf{S}(5,1)$ and $\mathbf{T}(7,-3)$.

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Key Concepts

Midpoint

The midpoint of a segment is the point that divides a line segment into two equal halves. It is calculated by taking the average of the corresponding coordinates of the endpoints. This concept is used in geometry to find a central point of a segment, which is particularly useful in various applications such as dividing a segment into equal parts or finding centers in geometric constructions.

Parametric Vector Equations for a Line

A parametric vector equation represents a line by expressing its points as the sum of a fixed point (often an endpoint) and a scalar multiple of a direction vector. This formulation is powerful because it allows us to describe the continuum of points along the line or segment using a parameter, typically varying between 0 and 1 when describing a finite segment.

Direction Vector

The direction vector is derived by subtracting the coordinates of the starting point from the coordinates of the terminal point of a segment. It indicates the direction and the relative magnitude along which the line extends. In the context of a parametric equation, the direction vector is scaled by a parameter to generate all points on the line or segment.

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