Problem 54 - University Calculus: Early Transcendentals

Question

Find the vector from the origin to the point of intersection of the medians of the triangle whose vertices are $$A(1,-1,2), \quad B(2,1,3), \quad \text { and } \quad C(-1,2,-1).$$

Step-by-Step Solution

Answer

The vector is \( \left( \frac{2}{3}, \frac{2}{3}, \frac{4}{3} \right) \).

Step 1: Calculate the Midpoints of the Sides

To find the point of intersection of the medians, we first need the midpoints of the sides opposite each vertex. The midpoint of side BC is calculated as follows: \[ M_{BC} = \left( \frac{2 + (-1)}{2}, \frac{1 + 2}{2}, \frac{3 + (-1)}{2} \right) = \left( \frac{1}{2}, \frac{3}{2}, 1 \right) \]Similarly, calculate the midpoints for the other sides using the formula for the midpoint \( (x_1+x_2)/2, (y_1+y_2)/2, (z_1+z_2)/2 \).

Step 2: Determine the Point of Intersection of the Medians (Centroid)

The centroid \(G\) of the triangle is found by taking the average of the coordinates of the vertices:\[ G = \left( \frac{1 + 2 + (-1)}{3}, \frac{-1 + 1 + 2}{3}, \frac{2 + 3 + (-1)}{3} \right) \]Simplifying each component:\[ G = \left( \frac{2}{3}, \frac{2}{3}, \frac{4}{3} \right) \]

Step 3: Write the Vector from the Origin to the Centroid

The vector from the origin \((0,0,0)\) to the centroid \((\frac{2}{3}, \frac{2}{3}, \frac{4}{3})\) is simply given by the coordinates of the centroid itself, since the vector points directly to it.Therefore, the vector is:\[ \mathbf{v} = \left( \frac{2}{3}, \frac{2}{3}, \frac{4}{3} \right) \]

Key Concepts

Midpoint Formula

The midpoint formula is an essential tool in coordinate geometry. It allows you to find the middle point of a line segment defined by two endpoints. This is useful for various applications, such as finding the centroid of a triangle. The formula is defined for three-dimensional space by averaging the corresponding coordinates of the endpoints.

Let's say you have two points in 3D space,

• Point 1: \( (x_1, y_1, z_1) \)
• Point 2: \( (x_2, y_2, z_2) \)

The midpoint \( M \) of the line segment connecting these two points is calculated as:\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)\]This results in a new point with coordinates that are the average of each pair of corresponding coordinates. The midpoint formula is indispensable for solving problems related to the intersection of medians in a triangle, as seen in the centroid finding process.

Vector Representation

Vectors are mathematical objects that have both magnitude and direction. They are crucial in coordinate geometry, helping to represent points or positions in space as well as physical quantities such as displacement, velocity, and force. When calculating the centroid of a triangle, vectors aid in understanding the direction from one point to another.

For a vector representation in 3D, consider the vector from the origin \( (0,0,0) \) to a point \( (x,y,z) \). This is simply the coordinates of the point itself, given by:\[\mathbf{v} = (x, y, z)\]Vectors are useful for describing locations relative to one another, particularly when determining the centroid of a triangle, which is the average of its vertices’ positions. By finding the vector from the origin to this centroid, one gains insight into the overall balance or center of mass of the triangle if it were physically represented.

Coordinate Geometry

Coordinate geometry, bridging algebra and geometry, uses algebraic equations to represent geometric properties. By assigning numerical coordinates to points in space, it turns geometric problems into algebraic ones, allowing for precise calculations and predictions.

The centroid of a triangle is a prime example of coordinate geometry's practical applications. By identifying the coordinates of the triangle’s vertices, one can apply arithmetic to find the centroid. As shown in the original solution, the centroid \( G \) of a triangle with vertices \( A(x_1, y_1, z_1), B(x_2, y_2, z_2), C(x_3, y_3, z_3) \) is given by averages:\[G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3}\right)\]This uses simple averages to derive a point effectively between the three vertices. Coordinate geometry simplifies solving for geometric constructs like centroids, medians, and vectors, making it a powerful tool in both mathematics and various science and engineering fields.