Given points A (0; 0), B (2; 2), C (5; -1). Find the dot product of the vector AC and CB.

Given points A (0; 0), B (2; 2), C (5; -1). Find the dot product of the vector AC and CB. Prove that triangle ABC is rectangular.

To find the coordinates of a vector, it is necessary to subtract the coordinates of the beginning from the coordinates of the end point of this vector:

AC = (5 – 0; -1 – 0) = (5; -1);

CB = (2 – 5; 2 + 1) = (-3; 3);

The scalar product of vectors is the sum of the products of their coordinates of the same name:

AC * CB = x1 * x2 + y1 * y2 = 5 * (-3) – 1 * 3 = -18;

To prove that a triangle is right-angled, it is necessary to find the lengths of all its sides and prove the Pythagorean theorem. Formula for finding the length of a segment:

d² = (x2 – x1) ² + (y2 – y1) ²;

AB² = (2 – 0) ² + (2 – 0) ²;

AB = √8;

BC² = (5 – 2) ² + (-1 -2) ²;

BC = √18;

CA² = (0 – 5) 2 + (0 – (-1)) 2;

CA = √26;

From the triangle ABC by the Pythagorean theorem:

CA² = AB² + BC²;

26 = 8 + 18;

26 = 26;

Q.E.D.