Question

Use vectors to show that the points form the vertices of a parallelogram. (1,1,3),(9,-1,-2),(11,2,-9),(3,4,-4)

Short answer

Yes, the four points indeed form a parallelogram.

Step-by-step solution

Form the vectors

First, form the four vectors AB, BC, CD, and DA from the four points (1,1,3), (9,-1,-2), (11,2,-9), and (3,4,-4) respectively.\n\nAB = B - A = (9,-1,-2) - (1,1,3) = (8,-2,-5)\nBC = C - B = (11,2,-9) - (9,-1,-2) = (2,3,-7)\nCD = D - C = (3,4,-4) - (11,2,-9) = (-8,2,5)\nDA = A - D = (1,1,3) - (3,4,-4) = (-2,-3,7)

Compare the vectors

Next, the vectors should be compared. Since it is known that in a parallelogram, opposite sides (or vectors in this context) are of equal length and direction, i.e., they are equal.\n\nComparing vectors AB and CD, they are equal but in opposite directions, as AB = -(CD) or (8,-2,-5) = -(-8,2,5).\n\nSimilarly, comparing vectors BC and DA, they are equal but in reverse directions, as BC = -(DA) or (2,3,-7) = -(-2,-3,7).

Conclude the result

As both pairs of vectors formed by the points are equal but in opposite directions, these four points form a parallelogram.

Key concepts

Vector Algebra

Vector algebra is an essential tool in physics and engineering, providing a way to represent quantities that have both magnitude and direction. It plays a crucial role in solving problems involving motion, force, and other vector quantities.

In the context of proving the existence of a parallelogram using vectors, the algebraic properties of vectors, such as addition, subtraction, and scalar multiplication, are used to derive and compare the characteristics of the vectors joining the vertices of the parallelogram.

For example, in the given exercise, the vectors AB, BC, CD, and DA represent the sides of a quadrilateral. By subtracting the coordinates of the starting point of a vector from its ending point, we obtain a new set of coordinates representing the vector's components in three-dimensional space. These components indicate the vector's direction and magnitude along the x, y, and z axes.

By applying vector algebra, specifically the concept of vector equality and multiplication by a scalar, we're able to determine that AB and CD have the same magnitude but opposite directions, as do BC and DA. This is represented mathematically by AB = -(CD) and BC = -(DA). Such relationships are pivotal in proving the presence of a parallelogram, as they indicate that opposite sides of the quadrilateral are parallel and equal in length, two defining properties of parallelograms.

Geometry in Space

Geometry in space, or spatial geometry, is the study of geometric figures in three dimensions. Unlike two-dimensional geometry, where we deal with shapes like circles and squares, three-dimensional geometry deals with cubes, spheres, and various polyhedra.

In our exercise, the vectors are embedded in three-dimensional space, which allows us to explore the geometric properties of the figure formed by the given points. Each vector can be seen as a directed line segment that has a defined position in space, representing the sides of a potential parallelogram.

The process begins by forming vectors that connect consecutive vertices. These vectors provide a representation of the sides of the quadrilateral in space. By using spatial reasoning, we can analyze the relationships between these vectors to conclude whether they form a parallelogram - a flat, two-dimensional figure with specific properties - within the three-dimensional space.

Understanding how to work with vectors in space is a foundational concept for many fields of study, including physics, engineering, and higher mathematics, as it forms the basis for analyzing forces, trajectories, and even the shapes and structures found in nature and man-made environments.

Parallelogram Properties

Parallelograms are four-sided shapes in plane geometry characterized by pairs of parallel and equal-length opposite sides. Their properties are a topic of discussion not only in geometry classes but also in vector analysis, where these properties can be demonstrated with vector algebra.

The defining attributes of a parallelogram include:

• Opposite sides that are equal in length
• Opposite sides that are parallel
• Opposite angles that are equal
• Diagonals that bisect each other

In our exercise, the use of vectors helps to show that the opposite sides of the quadrilateral are both parallel and equal in length. This is done by demonstrating that vector AB is parallel and equal in magnitude to vector CD but in the opposite direction, and similarly, vector BC is parallel and equal in magnitude to vector DA, completing the parallelogram.

The elegance of using vectors lies in their ability to encapsulate the necessary conditions for parallelism and equality through simple algebraic equations and comparisons. This approach to proving the properties of a parallelogram is particularly beneficial as it extends beyond the two-dimensional plane, allowing for a more comprehensive analysis in three-dimensional spaces.