Find the parametric equations of the line through the given...

Key Concepts

Vector Representation of Lines

Using vectors to represent lines simplifies the geometric interpretation by combining both a point and a direction. A line can be described as the set of points reached by adding a scalar multiple of the direction vector to a fixed point, thereby transitioning seamlessly from point to point along the line with the scalar controlling the position.

Parametric Equation of a Line

A parametric equation expresses the coordinates of points on a line as functions of a single parameter, typically denoted as t. This method involves starting from a fixed point on the line and adding multiples of a direction vector, resulting in expressions for each coordinate that describe the line in space.

Direction Vector

The direction vector provides the orientation of the line and is obtained by subtracting the coordinates of one given point from those of another. This calculation gives a vector that indicates the relative movement in each coordinate and is central to writing the parametric representation of the line.

Transcript

00:01 These problems, i ask us to find parametric equations of the line through the given pair of points.

00:09 And so i'm just going to do all the first four together in one video because they're all, you know, basically identical problems just with different values.

00:21 So we first basically, the first thing to say is none of these solutions are unique.

00:27 Okay.

00:30 There's there's an different number of ways to describe the same line in space so what i did is i always took the first point as the point that we go through when t equals zero and then took this this minus this as the vector pointing in the direction of the line.

00:57 Now i could have taken this as the point for t equals zero.

01:02 I could have taken this minus this as the vector.

01:05 So it's really because t is just a parameter, i could define, you know, i could put 2t in here or t minus three or you know t minus or t plus eight and then you know do some math and it's all going to represent the same line.

01:27 Now if it was a line segment, then you have to be careful.

01:30 But because t goes from minus infinity to infinity, you can always shift it and, you know, multiply it by some number and whatever, and you get the same line.

01:41 Your just, your origin will be different and your scale along the line would be different.

01:46 So these are not unique solutions.

01:49 None of these are.

01:51 And so you can get other solutions that are very basically equivalent, show the same line.

01:59 So in the first case, we have 1 minus 3.

02:04 So i took that as my first, my offset vector.

02:08 And then i took the direction as this minus this.

02:12 So the direct difference between the two points.

02:15 And that's minus 3, minus 7, minus 3.

02:18 And then we use our parameter t.

02:21 So what we can see here is that when t is zero we go through this point and when t is minus one we go through this point...