Question

Let \(P, Q,\) and \(R\) be the vertices of a parallelogram with adjacent sides \(P Q\) and \(P R\). In each case, find the other vertex \(S\). $$\text { a. } P(3,-1,-1), Q(1,-2,0), R(1,-1,2)$$ b. \(P(2,0,-1), Q(-2,4,1), R(3,-1,0)\)

Short answer

S is (3, 0, 1) for the first set, and (7, -5, -2) for the second set.

Step-by-step solution

Understanding the Parallelogram Property

A parallelogram has the property that its diagonals bisect each other. Hence, for a parallelogram with vertices \(P, Q, R,\) and \(S\), the midpoint of diagonal \(PR\) must coincide with the midpoint of diagonal \(QS\). We'll use this property to find \(S\).

Calculating Midpoint of PR

First, calculate the midpoint of \(P(3,-1,-1)\) and \(R(1,-1,2)\). Use the midpoint formula: \[ M_{PR} = \left( \frac{3+1}{2}, \frac{-1+(-1)}{2}, \frac{-1+2}{2} \right) = \left( 2, -1, \frac{1}{2} \right). \]

Calculating Midpoint of QS

Now, assume \(S(x,y,z)\) and set the midpoint of \(Q(1,-2,0)\) and \(S\) equal to the midpoint of \(PR\).\[ M_{QS} = \left( \frac{1+x}{2}, \frac{-2+y}{2}, \frac{0+z}{2} \right). \] This must equal \( \left( 2, -1, 0.5 \right) \).

Equating Midpoints to Find Coordinates of S

Set the coordinates of \(M_{QS}\) equal to \(M_{PR}\):1. \( \frac{1+x}{2} = 2 \) gives \( x = 3 \).2. \( \frac{-2+y}{2} = -1 \) gives \( y = 0 \).3. \( \frac{z}{2} = 0.5 \) gives \( z = 1 \).Thus, \( S = (3, 0, 1) \).

Validate the Solution

The midpoints \(M_{PR}\) and \(M_{QS}\) are both \( (2, -1, 0.5) \). Therefore, the solution \(S(3, 0, 1)\) maintains the parallelogram property.

Repeat Process for Second Set a

Repeat the steps for the second set of points:\(P(2,0,-1), Q(-2,4,1), R(3,-1,0)\).1. Midpoint \(M_{PR}\) of \(P\) and \(R\): \[ M_{PR} = \left( \frac{2+3}{2}, \frac{0+(-1)}{2}, \frac{-1+0}{2} \right) = \left( 2.5, -0.5, -0.5 \right). \]2. Assuming \(S(x, y, z)\) and using midpoint formula for \(Q\) and \(S\): \[ M_{QS} = \left( \frac{-2+x}{2}, \frac{4+y}{2}, \frac{1+z}{2} \right) = (2.5, -0.5, -0.5). \]3. Solving the equations: \( \frac{-2+x}{2} = 2.5 \) gives \( x = 7 \).\( \frac{4+y}{2} = -0.5 \) gives \( y = -5 \).\( \frac{1+z}{2} = -0.5 \) gives \( z = -2 \).Thus, \( S = (7, -5, -2) \).

Final Solution Validation

Checking midpoints for the second set ensures both are \( (2.5, -0.5, -0.5) \), confirming \(S(7, -5, -2)\) respects the parallelogram properties.

Key concepts

Coordinate Geometry

Coordinate geometry provides a framework for understanding geometric concepts using algebra. It helps in identifying and locating points in space via coordinates in three-dimensional space. A point in 3D is represented using three coordinates: \(x, y, z\). These coordinates define a point based on its position along the x, y, and z axes, respectively.
This geometrical framework is crucial for analyzing shapes like parallelograms in space. In the given problem, vertices of the parallelogram are given in the form of coordinates. Understanding their relative positions is necessary to apply further mathematical operations to solve for the missing vertex.
Coordinate geometry helps in visualizing how shapes like parallelograms behave, allowing for the application of the midpoint formula and vector mathematics efficiently in a structured manner.

Vector Mathematics

In the context of geometry, vectors are used to define positions, directions, and shapes in space. A vector connects two points in a coordinate system, characterized by its magnitude and direction. Vectors are depicted using coordinates, offering a clear view of spatial relations.
When working with parallelograms in 3D space, vector mathematics assists by representing sides and diagonals as vectors. For instance, a side from point P to Q can be seen as the vector \((Q_1 - P_1, Q_2 - P_2, Q_3 - P_3)\), where \(Q(x, y, z)\) and \(P(x, y, z)\) are the coordinates. This simplifies analysis and calculations, such as determining equal diagonals in a parallelogram, which is crucial for establishing the geometric properties and solving for unknown points like vertex S.

Midpoint Formula

The midpoint formula is an essential tool in geometry for computing the point that lies exactly halfway between two points in coordinate space. It is defined for two points \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\) by the formula:
\[ M = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2} \right) \]
This formula is crucial in solving the given exercise because it facilitates identifying the point that results from bisecting the line joining two vertices of a parallelogram. By ensuring the diagonals' midpoints coincide, we verify that we accurately locate vertex S.
This concept isn't limited to parallelograms, though; it serves a broader function in geometry to determine midpoints in various geometric shapes and forms, reinforcing the understanding of symmetry and balance in a geometric space.