How to write the correct formula for calculating the angle between vectors?

Question

Mathematics

Hello! The problem is this. I have a formula for calculating the angle between vectors. It is standard:

And now there are two vectors, the angle between which cannot be calculated in any way, since the right side of the equation is less than -1. These are the vectors:

var x1 = -0.045797169475341334, y1 = -0.9989507591808752;
var x2 = 0.04579716947534099, y2 = 0.9989507591808753;

As a result, the expression:

(x1 * x2 + y1 * y2) / Math.sqrt(Math.pow(x1, 2) + Math.pow(y1, 2)) * Math.sqrt(Math.pow(x2, 2) + Math.pow(y2, 2))

gives the result: -1.0000000000000002
And if we take the arccosine of this number, it will be NaN, which is understandable, since it is defined on the interval from -1 to 1.
How can I correct the formula so that this error does not occur?

Answer 1

var d = res < 0 ? Math.max(res, -1.0) : Math.min(res, 1.0);

Answer 2

you have the module at the top missing.

Answer 3

You think wrong .

var dot_product = x1 * y1 + x2 * y2;
var length_1 = Math.sqrt(x1 * x1 + y1 * y1);
var length_2 = Math.sqrt(x2 * x2 + y2 * y2);

var cos_a = dot_product / (length_1 * length_2);

Answer 4

You still think wrong
Must be

(x1 * x2 + y1 * y2) / Math.sqrt(Math.pow(x1, 2) + Math.pow(y1, 2)) / Math.sqrt(Math.pow(x2, 2) + Math.pow(y2, 2))