91Ó°ÊÓ

Vector addition is a fundamental operation that involves combining two or more vectors to produce a new vector. In the context of our problem, we need to add vectors \( \overline{\mathrm{OA}} \) and \( \overline{\mathrm{OB}} \).
Each vector has two components: an \( \mathbf{i} \) component (x-direction) and a \( \mathbf{j} \) component (y-direction). When adding vectors:

• Add the \( \mathbf{i} \) components together. For example, \( 4\mathbf{i} + 6\mathbf{i} = 10\mathbf{i} \).
• Add the \( \mathbf{j} \) components together. For example, \( 3\mathbf{j} - 1\mathbf{j} = 2\mathbf{j} \).

This method results in a new vector that represents the combined effect of the original vectors. In our problem, the sum is \( 10\mathbf{i} + 2\mathbf{j} \). This resultant vector becomes important when finding the centroid.

Unit vectors are essential in vector analysis because they serve as the building blocks for expressing any vector in a given coordinate system. A unit vector has a magnitude of one and entirely points in the direction of an axis.
The two most common unit vectors in a two-dimensional plane are:

• \( \mathbf{i} \) for the x-axis.
• \( \mathbf{j} \) for the y-axis.

By expressing a vector in terms of unit vectors, we distinctly specify how far to "walk" in each direction of the coordinate plane.
For instance, the vector \( 4\mathbf{i} + 3\mathbf{j} \) describes moving 4 units along the x-axis and 3 units along the y-axis from the origin. In our problem, finding \( \overline{\mathrm{OG}} \) involves expressing the centroid's position in terms of these unit vectors, leading to \( \frac{10}{3}\mathbf{i} + \frac{2}{3}\mathbf{j} \). This expression clearly denotes the centroid's location in the plane.

The geometric centroid, or simply the centroid, is the "center of mass" of a shape if the shape is made of a uniform material. In the triangle, the centroid \( G \) is the average position of all the points in all the coordinate directions of the triangle's vertices.
For a triangle formed by points (or vertices) A, B, and C, the centroid can be calculated using the formula:\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]where \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) are the vertices of the triangle. The centroid is often considered the "balance point" of the triangle.
In our exercise, vertices A, B, and O allow us to find the centroid \( \overline{\mathrm{OG}} \) by using vectors and confirming that the centroid divides each of the triangle's medians into a 2:1 ratio.
By adding the vector representations of these points \( \overline{\mathrm{OA}} \), \( \overline{\mathrm{OB}} \), and \( \overline{\mathrm{OC}} \) (where \( \overline{\mathrm{OC}} \) is zero), we easily apply the centroid formula: \( \overline{\mathrm{OG}} = \frac{(4\mathbf{i} + 3\mathbf{j}) + (6\mathbf{i} - \mathbf{j})}{3} \), leading to the solution.