Understanding displacements of points by interpreting directions

[onako]

Suppose that points x and y are given in Euclidean space. Point x is displaced to point x1 by

x1=x+a(x-y)

Given that a is positive number, how can it be shown that the distance x1 to y is larger than distance x to y. I'm mainly interested in a vector interpretation of the above update rule. In that sense, can (x-y) in the above rule be interpreted as
direction (force) from point y to point x? Similar interpretation is welcome.

Given negative a, the update is x1=x+a(y-x), so now the above intuition of a force is direction from point x to point y.

 

[tiny-tim]

hi onako!

… can (x-y) in the above rule be interpreted as
direction (force) from point y to point x?

yes, that's exactly correct …

x-y is the vector $\vec{YX}$, with magnitude |YX| and direction from Y to X

 

[onako]

So, adding a(x-y) to x, means that the distance x to y changes depending on a: positive a implies increased distance, and negative a implies decreased distance? It's a bit "non-rigid" to state that a vector is added to a point.

 

[tiny-tim]

So, adding a(x-y) to x, means that the distance x to y changes depending on a: positive a implies increased distance, and negative a implies decreased distance? It's a bit "non-rigid" to state that a vector is added to a point.

ah, no, it isn't added to a point, it's added to the vector $\vec{OX}$ …

$\vec{OX} + a\vec{XY} = \vec{OX_1}$ ​

draw the triangle, and you'll see why!

 

[blue_raver22]

I can provide a response to your question by explaining the vector interpretation of the given update rule.

First, let's define the vector a as a displacement vector that represents the distance and direction from point x to point y. This can be represented as a vector pointing from x to y, with a magnitude equal to the distance between the two points.

Now, in the update rule x1=x+a(x-y), the term (x-y) can be interpreted as a vector pointing from point y to point x. This vector represents the direction and magnitude of the displacement from point y to point x. This can be thought of as a force acting on point x, pulling it towards point y.

Since a is a positive number, it can be seen as a multiplier of the displacement vector (x-y). This means that the displacement of point x, x1, will be larger than the original distance between x and y, as it is being pulled by a greater force towards point y.

Similarly, in the case of negative a, the update rule x1=x+a(y-x) can be interpreted as a force acting on point x, pushing it towards point y. In this case, the displacement of point x, x1, will be smaller than the original distance between x and y, as it is being pushed away from point y.

In summary, the update rule can be interpreted as a force acting on point x, either pulling it towards or pushing it away from point y, resulting in a displacement to a new point x1. The magnitude of this displacement is dependent on the magnitude of the force, represented by the value of a.