Question

Find the area of the parallelogram defined by the given vectors. \(\vec{u}=\langle 1,2\rangle, \quad \vec{v}=\langle 2,1\rangle\)

Short answer

The area of the parallelogram is 3.

Step-by-step solution

Understand the Geometric Interpretation

A parallelogram's area can be found using the cross product of its defining vectors in two dimensions. The magnitude is the absolute value of the determinant formed by the components of the vectors.

Set Up the Determinant

To find the area, use the formula for the determinant of a 2x2 matrix formed by the vectors. Arrange the components of \(\vec{u}\) and \(\vec{v}\) into the matrix: \(\begin{bmatrix} 1 & 2 \ 2 & 1 \end{bmatrix}\).

Calculate the Determinant

The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is calculated as \(ad - bc\). For our vectors, this is \((1)(1) - (2)(2)\).

Compute the Result

Calculate the determinant: \(1 \cdot 1 - 2 \cdot 2 = 1 - 4 = -3\). The area of the parallelogram is the absolute value of the determinant, which is \(|-3| = 3\).

Key concepts

Parallelogram Area

The area of a parallelogram can be found using vectors, and it provides a useful way to understand the relationship between vector dimensions and real geometric figures. In vector geometry, the area of a parallelogram defined by two vectors is equivalent to the magnitude of their cross product. This area can also be considered as the product of the base of the parallelogram with the height perpendicular to that base. When dealing with two-dimensional vectors like \(\vec{u}=\langle 1,2\rangle\) and \(\vec{v}=\langle 2,1\rangle\), the area is derived using the determinant of a 2x2 matrix composed of the vector components.

By setting the vectors as rows or columns of a 2x2 matrix, the determinant gives a scalar value that represents twice the area of the triangle formed by the vectors. Therefore, the area of the parallelogram is simply the absolute value of this determinant. This method is efficient, requiring only simple arithmetic operations.

Cross Product

The cross product is a binary operation on two vectors in three-dimensional space, and it results in another vector. However, when dealing with vectors in two-dimensional space, like our example with \(\vec{u}=\langle 1,2\rangle\) and \(\vec{v}=\langle 2,1\rangle\), we adjust our understanding slightly. In 2D, the cross product is often represented by a scalar which is the determinant of the matrix formed by these vectors.

This determinant is calculated as: \(ad - bc\) for a matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\). For our vectors, forming the matrix \(\begin{bmatrix} 1 & 2 \ 2 & 1 \end{bmatrix}\), the cross product is the determinant \(1\cdot1 - 2\cdot2 = -3\). Even though it's negative, the area is always positive, which is why we take the absolute value, giving us an area of 3. This calculation illustrates the relationship of direction and magnitude in vector spaces.

Vector Geometry

Vector geometry involves using vectors to define and solve problems involving shapes and magnitudes in space. It helps us simplify complex two or three-dimensional problems by representing elements like points, lines, and planes as mathematical objects that can be manipulated algebraically.

Vectors such as \(\vec{u}=\langle 1,2\rangle\) and \(\vec{v}=\langle 2,1\rangle\) can describe directions and magnitudes. In the context of parallelograms, these vectors denote the sides, and by using operations like the cross product, we derive spatial properties like area. This geometric interpretation transforms abstract algebraic operations into a visual model that is easier to understand.

Vector geometry is crucial because it provides tools not only for calculating area but also for understanding perpendicularity, distances between points, and more. It serves as a bridge between algebra and geometry, offering a comprehensive framework for modeling and solving practical problems in fields such as physics, engineering, and computer graphics.