Question

In Exercises \(23-28,\) find a parametrization for the curve. the ray (half line) with initial point \((2,3)\) that passes through the point \((-1,-1)\)

Short answer

The parametrization for the given ray with initial point \((2,3)\) that passes through the point \((-1,-1)\) is \[r(t)= (2-3t, 3-4t)\] for \(t \geq 0\).

Step-by-step solution

Identify the Direction Vector

The first crucial step is to understand the direction of the ray. Since we know that the ray passes through the points \((2,3)\) and \((-1,-1)\), the direction vector of the ray can be calculated by taking the difference between the coordinates of these points, i.e., \[D = (-1 - 2, -1 - 3) = (-3, -4)\].

Construct the Parametrization of the Ray

The second step is to build the ray's parametrization using the initial point and the direction vector. The ray starts at a point and continues in the direction of a certain vector. This can be formulated as follows: For \(t \geq 0\), \[r(t) = P_0 + tD\], where \(P_0\) is the initial point and \(D\) is the direction vector. In this case, we have \[r(t) = (2,3) + t(-3,-4) = (2-3t, 3-4t)\].

Finalize the Parametrization of the Ray

To complete the parametrization, state that the parametrization for the given ray with initial point \((2,3)\) that passes through the point \((-1,-1)\) as \[r(t) = (2-3t, 3-4t)\] for \(t \geq 0\).