Question

Find the standard parametrization for the line segment joining the points below. Draw coordinate axes and sketch the segment, indicating the direction of increasing t for the parametrization (5,02) and (0,5,0) Find a parametrization for the line segment joining the wo points so that the paramatrization moves from (5,0,2) to (0,5,0) as t increases from 0 to 1 . (Type expressions using tas the variable.)

Answer 1

To determine the standard parametrization for the line that connects the points (5, 0, 2) and (0, 5, 0) we need to follow a few steps.

Let's denote the two points as P1 = (5, 0, 2) and P2 = (0, 5, 0).

1. First, we define the given points P1 and P2 as the start and end points respectively.

2. Then, we determine the standard parametrization for the line segment. This provides us with an expression that describes all the points on the line.

The parametrization R(t) of a line segment from point P1 to point P2 is given by the formula R(t) = (1 - t)P1 + tP2, where t is the parameter and 0 ≤ t ≤ 1.

If we apply this formula into our points, we will get a function where the input t will give us a point on the line. Here, when t = 0 we will get the point P1 and when t = 1 we get the point P2.

Therefore, the parametrization of the line segment will be as follows:

Parametrization R(t) = (1 - t) * (5, 0, 2) + t * (0, 5, 0)

The function obtained Parametrization R(t) will enable us to get any point on the line segment between (5, 0, 2) and (0, 5, 0) given a certain parameter t.

This parametrization allows us to transform the line segment into a parametric version, where t is the variable. As t increases from 0 to 1, the point moves from (5,0,2) to (0,5,0). This function will allow us to find out any intermediate point lying on the line segment.

That's how the standard line parametric method works. It's an easy method of finding all the points in line with the help of scalar t. This scalar t is the variable in these standard linear functions.

Answer 2

The standard parametrization for a line segment that runs from a point A to a point B is given by the function r(t) = A + t(B - A), where t is a variable that varies between 0 and 1.

In this case, the point A is (5, 0, 2) and point B is (0, 5, 0). Let's denote point A by its vector a = (5, 0, 2) and point B by its vector b = (0, 5, 0).

Therefore, to find the standard parametrization of the line segment from point A to point B, substitute points A and B into our standard parametrization function r(t) = A + t(B - A).

So, for the line segment between point A and point B, the parametrization function r(t) will be:

r(t) = (5, 0, 2) + t[(0, 5, 0) - (5, 0, 2)]
r(t) = (5, 0, 2) + t[-5, 5, -2]
r(t) = (5 - 5t, 0 + 5t, 2 - 2t)

This parametrization function describes every point between point A and point B depending on the value of the variable t.

When t = 0, we are at point A = (5, 0, 2).
As t increases, we move in a straight line from point A towards point B.
When t = 1, we have reached point B = (0, 5, 0).

So this parametrization function accurately describes the line segment from point A (5,0,2) to point B (0,5,0), with the motion along the line from point A to point B as t varies from

0 to 1. In a graph, the direction of increasing t would be from point A towards point B.