Find the closest point or points to 𝐛=(-1,2)^T that lie on (a) the x-axis, (b) the y-axis, (c) t

Find the closest point or points to 𝐛=(-1,2)^T that lie on...

Key Concepts

Euclidean Distance

Euclidean distance is a measure of the straight?line distance between two points in a Euclidean space. In the context of locating the point on a line that is closest to a given point, it provides the criterion that we seek to minimize, ensuring the shortest possible connection between the point and the line.

Orthogonal Projection

Orthogonal projection involves dropping a perpendicular from the given point to the line or subspace. The point where this perpendicular meets the line is the closest point on the line to the given point, making orthogonal projection a fundamental concept in solving such minimization problems.

Distance Minimization

Distance minimization is the underlying principle used to find the closest point on a set or line to a given point. This typically involves setting up a function that represents the distance between the point and an arbitrary point on the line, and then finding its minimum value, often using geometric observations such as orthogonality to simplify the process.

Analytic Geometry of Lines

Analytic geometry of lines involves the representation and manipulation of lines using algebraic equations. Understanding different forms of line equations (such as slope-intercept and standard forms) is crucial for setting up the projection and minimization problems required to find the closest points on various lines.

Transcript

00:02 All right, so we want to find the point on the line, x over a plus y over b, that's closest to the origin.

00:17 So let's see here.

00:20 If we solve for y, we write this as a function of y, well, this is going to be negative b over a, x plus b.

00:40 And now i guess we're assuming here that no yeah that's it so y is equal to this which uh let's see we'll just leave like that okay so the distance from the origin so we'll call that d as a function of x is going to be the square root of x squared plus y squared but we have this expression for y squared so x squared plus negative b over a x plus b squared okay and x what's the domain of x well x can be anything right because the domain of this line is all real numbers so x is just any real number so let's notice first of all that as x goes to plus or minus infinity well this is going to be behaving like a linear function right but positive so we're going to have we can basically factor out in x squared and it's going to be an absolute value of x.

02:09 So d is going to go to infinity.

02:11 I mean, that makes sense.

02:12 Like as x gets far away from zero, of course the distance is going to be far away from zero.

02:19 Okay, so let's take the derivative, and hopefully we'll just have one critical point, and we'll know that has to be the minimum.

02:27 Okay, so this is 1 over 2 square root x squared plus negative b over a, x plus b square and then chain roll take the derivative of what's inside the square root and that's going to be 2x plus 2 times negative b over a plus b okay and if we want d prime to be 0 x there negative b over a x plus b and then we have another chain rule so and then times so x goes to plus or minus infinity, d goes to infinity...

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