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Chapter 10: Problem 8

(a) The centroid of a triangle lies on the line segment connecting any one of the three vertices of the triangle with the midpoint of the opposite side. Its location on this line segment is two-thirds of the distance from the vertex. If the three vertices are given by the vectors \(v_{1}, v_{2},\) and \(v_{3},\) write the centroid as a convex combination of these three vectors. (b) Use your result in part (a) to find the vector defining the centroid of the triangle with the three vertices \(\left[\begin{array}{l}2 \\\ 3\end{array}\right],\left[\begin{array}{l}5 \\ 2\end{array}\right],\) and \(\left[\begin{array}{l}1 \\ 1\end{array}\right]\).

Short Answer

Expert verified
The centroid for the given vertices is \( \begin{bmatrix} \frac{8}{3} \\ 2 \end{bmatrix} \).

Step by step solution

01

Understanding the Problem

We need to find an expression for the centroid (G) of a triangle using the vectors \(v_{1}, v_{2}, \) and \(v_{3}\) representing the vertices of a triangle.
02

Formula for Centroid as a Convex Combination

The centroid of a triangle can be expressed as the average of the position vectors of its vertices. Therefore, the centroid \(G\) is given by the formula: \[ G = \frac{1}{3}(v_1 + v_2 + v_3) \]. This formula is a convex combination of the vectors \(v_{1}, v_{2}, \) and \(v_{3}\).
03

Identifying Given Vectors

The given vectors for the vertices of the triangle are \(v_{1} = \begin{bmatrix} 2 \ 3 \end{bmatrix}, v_{2} = \begin{bmatrix} 5 \ 2 \end{bmatrix}, v_{3} = \begin{bmatrix} 1 \ 1 \end{bmatrix}\).
04

Applying the Formula to Given Vectors

Substitute the given vectors into the formula: \[ G = \frac{1}{3} \left( \begin{bmatrix} 2 \ 3 \end{bmatrix} + \begin{bmatrix} 5 \ 2 \end{bmatrix} + \begin{bmatrix} 1 \ 1 \end{bmatrix} \right) \].
05

Calculate the Sum of Vectors

Add the vectors: \( \begin{bmatrix} 2 \ 3 \end{bmatrix} + \begin{bmatrix} 5 \ 2 \end{bmatrix} + \begin{bmatrix} 1 \ 1 \end{bmatrix} = \begin{bmatrix} 8 \ 6 \end{bmatrix} \). This results from adding the components vertically.
06

Calculate the Centroid Vector

Now, divide the resulting vector by 3: \( G = \frac{1}{3} \begin{bmatrix} 8 \ 6 \end{bmatrix} = \begin{bmatrix} \frac{8}{3} \ 2 \end{bmatrix} \). This is done by dividing each component of the vector by 3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convex Combination
A convex combination is a special way to combine several vectors using weights. The key properties are that the weights must add up to 1, and each weight must be a non-negative number. In the context of a triangle, when we talk about the centroid, it can be defined as a convex combination of the vectors representing its vertices.For three vertices given by vectors \( v_1, v_2, \) and \( v_3 \), the centroid \( G \) is calculated using the formula:
  • \( G = \frac{1}{3}(v_1 + v_2 + v_3) \)
This means each vector contributes equally to the centroid, with a weight of \( \frac{1}{3} \) for each vector. The centroid thus represents a balance point, where all sides and orientations of the triangle contribute equally.
Vector Addition
Vector addition is a fundamental operation in mathematics, especially in cases like finding centroids. Adding vectors means that we combine them to form a new vector. Each component of the vectors is added separately.Suppose we have three vectors from the vertices of a triangle:
  • \( v_1 = \begin{bmatrix} 2 \ 3 \end{bmatrix} \)
  • \( v_2 = \begin{bmatrix} 5 \ 2 \end{bmatrix} \)
  • \( v_3 = \begin{bmatrix} 1 \ 1 \end{bmatrix} \)
When we perform vector addition on them, we calculate:
  • \( v_1 + v_2 + v_3 = \begin{bmatrix} 2 \ 3 \end{bmatrix} + \begin{bmatrix} 5 \ 2 \end{bmatrix} + \begin{bmatrix} 1 \ 1 \end{bmatrix} = \begin{bmatrix} 8 \ 6 \end{bmatrix} \)
Each horizontal and vertical component from the vectors is summed up individually to form the resultant vector.
Position Vectors
Position vectors are vectors that represent the position or location of a point relative to an origin. In geometric contexts, especially triangles, position vectors represent the vertices by indicating where they are located concerning some fixed origin.Think of a position vector as an arrow pointing from the origin to a specific vertex of the triangle. Given position vectors:
  • \( v_1 = \begin{bmatrix} 2 \ 3 \end{bmatrix} \)
  • \( v_2 = \begin{bmatrix} 5 \ 2 \end{bmatrix} \)
  • \( v_3 = \begin{bmatrix} 1 \ 1 \end{bmatrix} \)
These vectors help us determine not only the centroid but also provide essential data on the shape and orientation of the triangle between these three points. They spell out the direction and relative distance from the origin to each of the vertices.
Midpoint
The midpoint in a geometric sense is the point that is exactly halfway along the line segment connecting two points. If the endpoints of the segment are \( A \) and \( B \), the midpoint \( M \) can be found using the position vectors of those points.For a line segment with endpoints given by position vectors \( v_1 \) and \( v_2 \), the midpoint \( M \) is calculated as follows:
  • \( M = \frac{1}{2} (v_1 + v_2) \)
This is derived by taking the average of the vectors, which involves adding them and then dividing by 2. This concept is crucial in finding the centroid because the centroid divides the segment from a vertex to the midpoint of the opposite side in a 2:1 ratio.

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