How to find the angle between vectors

How to find the angle between vectors?

The angle between the two vectors that go withone point is the nearest angle, a rotation on which, the first vector around its origin, will lead it to the position of the second vector. But how to find the angle between the vectors? Our article will tell about this.

We have two nonzero vectors that come from a single point - a vector A having coordinates (x1, y1), the vector B having the coordinates (x2, y2). The angle between them is μ.

  1. We use the definition of a scalar product to find the degree measure of the angle μ. We obtain (A, B) = | A | * | B | * cos (μ). We express the cosine of the angle. And so cos (μ) = (A, B) / (| A | * | B |).
  2. It can also be found by the formula: (A, B) = x1* x2+ y1* y2. When the scalar product of the vectors iszero - the vectors are perpendicular (the angle between them is 90 °), so further calculations are not performed. If the scalar product has a positive sign, the angle between the vectors is acute, if the negative sign is an obtuse angle.
  3. Further, we assume that the lengths of the vectors A and B are given by the formulas | A | = v (x1² + y1²), | B | = v (x2² + y2²). The lengths of the vectors are the square roots of the sums of the squares of their coordinates.
  4. The values ​​of the lengths of the vectors and the scalar product found by you are substituted into the formula obtained from step 2, which will allow us to find the cosine of the angle. We have: cos (μ) = (x1* x2+ y1* y2) / (v (x1² + y1²) + v (x2² + y2²)).
  5. Having the cosine value to find the angle betweenvectors av, we use the Bradys table. Also for this you can take the arccosine. Then we obtain μ = arccos (cos (μ)). The Bradys table can be viewed, for example, here: www.math.com.ua.

To find the angle between the vectors online, you can use, for example, such links: www.ru.onlinemschool.com and www.mathserfer.com.

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