find the distance from the point to the line. (0,0,0) ; x=5+3 t, y=5+4 t, z=-3-5 t

Name: Problem 34, Chapter 12: Thomas Calculus
Uploaded: 2019-08-02T22:44:13-08:00
Duration: 4 min 36 s
Channel: Tyler Gaona
Description: find the distance from the point to the line. (0,0,0) ; x=5+3 t, y=5+4 t, z=-3-5 t

Find the distance from the point to the line. (0,0,0) ;...

Key Concepts

Vector Representation of a Line

In three-dimensional space, a line can be represented in vector form by specifying a point on the line and a direction vector. The parametric equations of the line express each coordinate as a sum of the corresponding coordinate of a fixed point and a scaled version of the direction component, which together capture the line’s position and orientation in space.

Direction Vector

The direction vector of a line is derived from the coefficients of the parameter in the line's parametric equations. It indicates the line’s orientation in three-dimensional space and is essential for operations that involve determining alignment, such as calculating distances between geometric objects.

Cross Product in Three Dimensions

The cross product of two vectors in three dimensions yields a new vector that is perpendicular to both original vectors. This operation is critical when determining the shortest distance from a point to a line, as it helps isolate the component of a vector that is perpendicular to the line's direction.

Distance from a Point to a Line

The distance from a point to a line in space is defined as the length of the perpendicular segment from the point to the line. This distance can be calculated by taking the vector from the point to any point on the line, computing its cross product with the line's direction vector, and then dividing the magnitude of the resulting vector by the magnitude of the direction vector. This method leverages vector operations to quantify the shortest separation between the point and the line.

Transcript

00:01 So here we're given this point, which in this problem is the origin, and we're asked to find the distance of this point from this point to the line given by these equations.

00:14 And so first let's just try and draw a general picture of what's going on.

00:18 So let's say here's the line and here's the point.

00:23 Okay, so let's call this p.

00:27 And let's say we can just find another point on the line, call it s.

00:30 Okay, now it should be sort of clear from the picture that the point where this line, the distance from this point to this line is the smallest, is when you can imagine like taking this point s, sliding it along the line, when this length of this segment is going to be the shortest is when it meets the line at a right angle.

01:09 Okay, so right now the distance is going to be given by the length of ps and then assuming we know the angle theta between them, cosine theta.

01:23 That's just from the fact that this is a right triangle.

01:28 But now we might think, well, maybe i don't want to find the angle between the line and this segment from the line to the point.

01:40 So what i want to do in that case is there's a little trick here.

01:52 I'm sorry, and saying cosine, that's wrong.

01:56 So cosine would be adjacent over hypotenuse.

01:59 What i meant, what i really want is this guy right here, which is opposite over hypotenuse.

02:05 So that's sine theta.

02:07 Sorry about that.

02:09 But still, i don't necessarily know this angle.

02:13 But what i can do is just say that...