Question

write a parametric vector equation and a system of parametric Cartesian equations for the line containing the given points $S$ and $T$. $\mathbf{S}(2 a, b), \mathbf{T}(3 a, 2 b)$

   write a parametric vector equation and a system of parametric Cartesian equations for the line containing the given points $S$ and $T$.
$\mathbf{S}(2 a, b), \mathbf{T}(3 a, 2 b)$

Modern Analytic Geometry

Modern Analytic Geometry

William Wooton,… 1st Edition

Chapter 2, Problem 10

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We have \( S(2a, b) \) and \( T(3a, 2b) \). Show more…

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write a parametric vector equation and a system of parametric Cartesian equations for the line containing the given points $S$ and $T$. $\mathbf{S}(2 a, b), \mathbf{T}(3 a, 2 b)$

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Key Concepts

Parametric Vector Equation

A parametric vector equation of a line represents the line as the sum of a fixed point (position vector) and a scalar multiple of a direction vector. It is generally written as r(t) = r0 + t*v, where r0 is any point on the line and v indicates the direction of the line.

Direction Vector

The direction vector is crucial for defining the line's orientation and is commonly obtained by subtracting the coordinates of two different points on the line. It shows how the line moves in space as the parameter changes, directly affecting the formulation of the line’s equations.

Parametric Cartesian Equations

Parametric Cartesian equations break down the vector equation into separate equations for each coordinate variable, typically in the form x = x0 + t*v_x and y = y0 + t*v_y. These equations express each coordinate as a function of a common parameter, providing an alternative representation of the line that can be useful for solving geometric and analytic problems.

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