Question

Find the area of the parallelogram defined by the given vectors. \(\vec{u}=\langle 2,0\rangle, \quad \vec{v}=\langle 0,3\rangle\)

Short answer

The area of the parallelogram is 6.

Step-by-step solution

Understand the Concept of Area for a Parallelogram using Vectors

The area of a parallelogram defined by two vectors \(\vec{u}\) and \(\vec{v}\) is given by the magnitude of their cross product. In two dimensions, for vectors \(\vec{u} = \langle u_1, u_2 \rangle\) and \(\vec{v} = \langle v_1, v_2 \rangle\), the magnitude of the cross product simplifies to \(|u_1 \cdot v_2 - u_2 \cdot v_1|\).

Identify the Components of the Vectors

Given vectors are \(\vec{u} = \langle 2, 0 \rangle\) and \(\vec{v} = \langle 0, 3 \rangle\). Here, \(u_1 = 2\), \(u_2 = 0\), \(v_1 = 0\), and \(v_2 = 3\).

Calculate the Determinant for the Area

Substitute the components into the determinant formula: \(|u_1 \cdot v_2 - u_2 \cdot v_1| = |2 \cdot 3 - 0 \cdot 0| = |6| = 6\).

Conclusion

The area of the parallelogram is the absolute value of the determinant calculated in Step 3, which is \(6\).

Key concepts

Area of Parallelogram

The area of a parallelogram in a vector context is a geometrical representation of how much space is enclosed within its edges. Imagine two vectors, \(\vec{u}\) and \(\vec{v}\), originating from the same point. They form two sides of the parallelogram. The base is one vector, and the side slanting outward is formed by the second vector.
The general formula for calculating this area is obtained from the magnitude of their cross product. In simpler terms, you compute how far apart these vectors can stretch out from each other. In two dimensions, this is particularly straightforward. The vectors lie on a plane, and their cross product is a scalar, which simplifies the calculation.
Interestingly, this area calculation involves determinants, a unique aspect of linear algebra that ties deeply with spatial geometry. This application finds practical uses in fields such as physics and engineering, where area determination between vectors is crucial.

Cross Product

The cross product is a mathematical operation that takes two vectors and yields another vector perpendicular to both. However, in two dimensions, we use a similar concept where it results in a scalar. For vectors \(\vec{u} = \langle u_1, u_2 \rangle\) and \(\vec{v} = \langle v_1, v_2 \rangle\), the simplified version of cross product magnitude is \(|u_1 \cdot v_2 - u_2 \cdot v_1|\).
This scalar represents the area of the parallelogram spanned by the two vectors. It essentially tells us how much one vector is slanting away from another, creating the sense of area. This method is particularly useful in two-dimensional space where direct vector cross products do not apply, yet their magnitude matters.
Understanding this helps in visualizing vector fields and surfaces, offering an intuitive grasp of space and plane how they interact. This concept lays the groundwork for more complex vector operations and calculations.

Determinant

A determinant is a special number calculated from a matrix. It holds information about the matrix’s properties, such as whether it's invertible or how it scales areas and volumes. For a 2x2 matrix formed by two vectors \(\vec{u}\) and \(\vec{v}\) given as \(\begin{bmatrix} u_1 & v_1 \ u_2 & v_2 \end{bmatrix}\), its determinant is \(u_1 \cdot v_2 - u_2 \cdot v_1\).
The absolute value of this determinant provides the area of the parallelogram defined by the vectors. This simple number carries profound geometric meaning. It signifies how the vectors stretch and rotate the space around them. It can be thought of as a scaling factor that tells us how much a shape will grow or shrink.
Determinants find their applications in numerous areas such as systems of linear equations, transformations in graphics, and more. The elegance of determinants transforms complex calculations into simpler forms, fostering greater comprehension in the study of linear algebra.