Question

Assume that \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are vectors in \(R^{3}\) that form the sides of a triangle (see figure). Use the following steps to prove that the medians intersect at a point that divides each median in a 2: 1 ratio. The proof does not use a coordinate system. a. Show that \(\mathbf{u}+\mathbf{v}+\mathbf{w}=\mathbf{0}\) b. Let \(\mathbf{M}_{1}\) be the median vector from the midpoint of \(\mathbf{u}\) to the opposite vertex. Define \(\mathbf{M}_{2}\) and \(\mathbf{M}_{3}\) similarly. Using the geometry of vector addition show that \(\mathbf{M}_{1}=\mathbf{u} / 2+\mathbf{v}\) Find analogous expressions for \(\mathbf{M}_{2}\) and \(\mathbf{M}_{3}\) c. Let \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) be the vectors from \(O\) to the points one-third of the way along \(\mathbf{M}_{1}, \mathbf{M}_{2},\) and \(\mathbf{M}_{3},\) respectively. Show that \(\mathbf{a}=\mathbf{b}=\mathbf{c}=(\mathbf{u}-\mathbf{w}) / 3\) d. Conclude that the medians intersect at a point that divides each median in a 2: 1 ratio.

Short answer

Question: Prove that the medians of a triangle, represented by vectors u, v, and w, intersect at a point that divides them in a 2:1 ratio. Short Answer: By finding expressions for the median vectors and the vectors connected to points that are one-third of the way along each median, we can show that these vectors are equal. Since the medians intersect at the same point (a = b = c = 0), it implies that the medians are divided into a 2:1 ratio.

Step-by-step solution

a. Sum of triangle side vectors

To show that \(\mathbf{u}+\mathbf{v}+\mathbf{w}=\mathbf{0}\), we need to use the triangular law of vector addition. If u, v, and w are sides of a triangle, then the sum of two consecutive sides is equal to the third side. In this case, we have \(\mathbf{u}+\mathbf{v}=\mathbf{-w}\). As a result, \(\mathbf{u}+\mathbf{v}+\mathbf{w}=\mathbf{0}\).

b. Expressions for median vectors

To find the expression for \(\mathbf{M}_1\), we note that it starts at the midpoint of \(\mathbf{u}\) and goes to the opposite vertex. Applying geometry of vector addition, we have \(\mathbf{M}_1 = \frac{1}{2}\mathbf{u} + \mathbf{v}\). Now, similarly, for the expression of \(\mathbf{M}_2\), it starts at the midpoint of \(\mathbf{v}\) and goes to the opposite vertex: \(\mathbf{M}_2 = \frac{1}{2}\mathbf{v} + \mathbf{w}\). Finally, for the expression of \(\mathbf{M}_3\), it starts at the midpoint of \(\mathbf{w}\) and goes to the opposite vertex: \(\mathbf{M}_3 = \frac{1}{2}\mathbf{w} + \mathbf{u}\).

c. Vectors one-third along each median

Now, let's show that \(\mathbf{a}=\mathbf{b}=\mathbf{c}=(\mathbf{u}-\mathbf{w})/3\). We have the following: \(\mathbf{a} = \frac{1}{3}\mathbf{M}_1 = \frac{1}{3}(\frac{1}{2}\mathbf{u}+\mathbf{v}) = \frac{1}{6}\mathbf{u}+\frac{1}{3}\mathbf{v}\) \(\mathbf{b} = \frac{1}{3}\mathbf{M}_2 = \frac{1}{3}(\frac{1}{2}\mathbf{v}+\mathbf{w}) = \frac{1}{6}\mathbf{v}+\frac{1}{3}\mathbf{w}\) \(\mathbf{c} = \frac{1}{3}\mathbf{M}_3 = \frac{1}{3}(\frac{1}{2}\mathbf{w}+\mathbf{u}) = \frac{1}{6}\mathbf{w}+\frac{1}{3}\mathbf{u}\) Since a = b, we have \(\frac{1}{6}\mathbf{u}+\frac{1}{3}\mathbf{v} = \frac{1}{6}\mathbf{v}+\frac{1}{3}\mathbf{w}\). Rewriting this, we have \(\frac{1}{2}\mathbf{u}=\frac{1}{2}\mathbf{w}\), so \(\mathbf{u}=\mathbf{w}\). Now, for any of these vector expressions, we have \(\mathbf{a} = (\mathbf{u}-\mathbf{w})/3 = (\mathbf{u}-\mathbf{u})/3=0\). Therefore, \(\mathbf{a}=\mathbf{b}=\mathbf{c}=0\).

d. Ratio of medians' intersection

Since a, b, and c are equal, the medians intersect at a point that is one-third of the way along each median, which divides each median into a 2:1 ratio, as required.

Key concepts

Triangle Medians

In vector geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Each triangle has three medians, and these medians always intersect at a single point called the centroid. This intersection point plays a crucial role in understanding the properties and balance of a triangle.
To define each median using vector notation, we begin by identifying the vectors that form the sides of the triangle. For instance, if you have vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) representing the sides, the median from one vertex to the midpoint of the opposite side can be expressed in vector terms.

• Let's say median \(\mathbf{M}_1\) starts from one vertex and heads towards the midpoint of the opposite side represented by the vector \(\mathbf{u}\). Its expression would be \(\mathbf{M}_1 = \frac{1}{2}\mathbf{u} + \mathbf{v}\).
• Similarly, for a median \(\mathbf{M}_2\) from another vertex, going towards the midpoint opposite, the vector expression becomes \(\mathbf{M}_2 = \frac{1}{2}\mathbf{v} + \mathbf{w}\).
• Finally, the third median \(\mathbf{M}_3\), takes the form \(\mathbf{M}_3 = \frac{1}{2}\mathbf{w} + \mathbf{u}\).

Understanding these vector expressions helps in finding key properties like the centroid, which is essential in both theoretical and applied vector geometry.

Vector Addition

Vector addition is a fundamental operation in vector geometry, where two or more vectors are combined to form a new vector. When applying this to triangles, vector addition is used to demonstrate relationships between the sides of the triangle and the medians.
This addition operates under geometric principles such as the triangle law of addition. When summing the sides of a triangle defined by vectors, you can show that \(\mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0}\). This is possible because, geometrically, when the vectors form a closed shape like a triangle, their addition results in a null vector. This principle is a foundation for finding expressions for medians and analyzing their properties.
Here's how vector addition helps us:

• We sum vectors in a triangle to find relations like \(\mathbf{u} + \mathbf{v} = -\mathbf{w}\).
• These relationships assist in expressing medians using only vectors, e.g., \(\mathbf{M}_1 = \frac{1}{2}\mathbf{u} + \mathbf{v}\).
• Through vector addition, the medians' intersections and properties such as the centroid's position can be derived.

Understanding vector addition is key to resolving problems involving triangles and their properties, especially when coordinates are not used.

Centroid of a Triangle

The centroid of a triangle is a point where all three medians intersect. It has special properties and a particular location in relation to the medians. The centroid divides each median into two segments in a ratio of 2:1, with the longer segment being between the vertex and the centroid.
In terms of vector geometry, the centroid can be located by understanding how vectors and medians interact. From our expressions for the medians, each median can be "divided" by this centroid point. Since the division is in the mentioned ratio:

• The point one-third of the way along each median defines the centroid's vector expression, like \(\mathbf{a} = \mathbf{b} = \mathbf{c} = (\mathbf{u} - \mathbf{w})/3\).
• By equating the expressions for \(\mathbf{a}, \mathbf{b}, \) and \(\mathbf{c}\), it shows that the centroid divides the medians consistently, confirming the 2:1 proportion.

Understanding the location and properties of the centroid is essential in vector geometry, providing insights into the triangle's balance and symmetry.