91Ó°ÊÓ

A direction vector is a vital component when dealing with parametric equations for a line or ray. It essentially tells us the direction in which the line or ray extends.
For instance, consider trying to define a ray (or half-line) through known points. Here, you start by identifying two points along this ray: the "initial point" and another point that lies on the ray.

• To find the direction: Subtract the coordinates of the initial point from the coordinates of the other point.

In our exercise, the direction vector was found by subtracting the coordinates of the initial point \(2, 3\) from the coordinates of the point \(-1,-1\), yielding a direction vector of \(-3, -4\).
This means that as you move along the ray, each step in one direction is met with a proportional change in the other direction as described by this vector.

The initial point serves as the starting point for a ray. It is the point from which you begin to measure distance and direction defined by the direction vector.
This idea is simple to grasp: just think of it as the anchor of your ray. In our example, this is the point \(2, 3\).

• When dealing with parametric equations, the initial point provides a base location.
• It helps in constructing the parametric equations by forming part of the equations themselves.

Using the initial point along with a direction vector in parametric equations allows you to create a set of coordinates that the ray will pass through. In other words, without an initial point, it would be impossible to define where the ray begins, thus making direction over distance meaningless.

Parametrization is the mathematical way to describe the coordinates of a ray in terms of a variable or parameter. This variable, often represented by \(\lambda\), acts like a scale factor that "moves" along the ray's direction vector.

• In this context, \(\lambda\) is a non-negative scalar, used to ensure the ray extends in the correct direction away from the initial point.

The general formula for such a parametrization is built on the direction vector and the initial point:

1. Start with a base as the initial point.
2. Add the direction vector, scaled by \(\lambda\), to grow points along the ray.

So, for our exercise scenario:- The initial point is \(2,3\).- The direction vector is \(-3, -4\).The parametrized equations become \[x(\lambda) = 2 - 3\lambda\] \[y(\lambda) = 3 - 4\lambda\] where \(\lambda \geq 0\).
This ensures that the value of \(\lambda\) allows you to generate coordinates only in the forward direction of the ray, starting from its initial point.