Triangle ABC is an equilateral triangle, with O its centroid
Prove Vector OA + vector OB + vector OC = Vector 0
a few seconds with google produced this link.
https://math.stackexchange.com/questions/2266172/does-the-sum-of-three-vectors-originating-from-the-centroid-of-a-triangle-and-po
To prove that vector OA + vector OB + vector OC = vector 0, we can use the properties of an equilateral triangle and the definition of a centroid.
First, let's establish some notation. Let A, B, and C denote the vertices of the equilateral triangle, and let O represent its centroid. We will also denote the vectors using boldface letters, such as vector OA or vector OB.
To prove the given statement, we need to show that the sum of vectors OA, OB, and OC equals the zero vector.
1. We know that in an equilateral triangle, all three sides are equal in length. Therefore, we can say that vector AB = vector BC = vector CA.
2. Now, consider the centroid O. It is a point that is located two-thirds of the way from each vertex to the opposite side. In other words, vector OA = vector OB = vector OC.
3. Using the information from step 1 and 2, we can rewrite the equation as vector OA + vector OB + vector OC = vector OA + vector OA + vector OA.
4. Applying the properties of vector addition, we can combine the three vectors: vector OA + vector OA + vector OA = 3 * vector OA.
5. Since vector OA = vector OB = vector OC, we can substitute vector OA with vector OB or vector OC in step 4: 3 * vector OA = 3 * vector OB = 3 * vector OC.
6. Finally, we can use another property of vector addition: a scalar multiple of any vector is equal to the zero vector if and only if the scalar is zero. In this case, since 3 is not zero, we can conclude that 3 * vector OA = 3 * vector OB = 3 * vector OC cannot be equal to the zero vector.
Therefore, it is clear that vector OA + vector OB + vector OC does not equal vector 0.