Question

The vectors \(\mathbf{A}=a \mathbf{i}+b \mathbf{j}\) and \(\mathbf{B}=\boldsymbol{c} \mathbf{i}+d \mathrm{j}\) form two sides of a parallelogram. Show that the area of the parallelogram is given by the absolutc valuc of the determinant $$ \left|\begin{array}{ll} a & b \\ c & d \end{array}\right| $$ (Also see Chapter 6 , Section 3.)

Short answer

Area = \bigg| \begin{vmatrix} a & b \ c & d \ \bigg| \.

Step-by-step solution

Understand the Problem

Identify that the problem involves calculating the area of a parallelogram formed by two vectors \(\textbf{A} = a \textbf{i} + b \textbf{j}\) and \(\textbf{B} = c \textbf{i} + d \textbf{j}\). The goal is to show that the area is given by the absolute value of the determinant of a 2x2 matrix.

Express the Vectors in Component Form

Write the vectors \(\textbf{A}\) and \(\textbf{B}\) in component form as: \[ \textbf{A} = \begin{pmatrix} a \ b \ \textbf{B} = \begin{pmatrix} c \ d \ \]

Recall the Formula for the Area of a Parallelogram

The area \(A\) of a parallelogram formed by vectors \(\textbf{A}\) and \(\textbf{B}\) is given by the magnitude of the cross product of the vectors.

Use Cross Product for Two-Dimensional Vectors

For two-dimensional vectors \(\textbf{A}\) and \(\textbf{B}\), the cross product is given by the determinant of the matrix formed by the components of the vectors: \[ \textbf{A} \times \textbf{B} = \begin{vmatrix} a & b \ c & d \ \] This determinant is computed as \(ad - bc\)

Compute the Area

The area of the parallelogram is the absolute value of the determinant: \[ \text{Area} = | \textbf{A} \times \textbf{B} | = | ad - bc | \]

Conclusion

Thus, we've shown that the area of the parallelogram formed by vectors \(\textbf{A}=a \textbf{i}+b \textbf{j}\) and \(\textbf{B}=c \textbf{i}+d \textbf{j}\) is given by \[ \text{Area} = \bigg| \begin{vmatrix} a & b \ c & d \ \bigg| \]

Key concepts

Parallelogram Area

The area of a parallelogram can be found using vectors that form its sides. If you consider two vectors, \(\textbf{A} = a \textbf{i} + b \textbf{j}\) and \(\textbf{B} = c \textbf{i} + d \textbf{j}\), these vectors represent two adjacent sides of the parallelogram. To find the area, you use the cross product of these vectors. The magnitude of this cross product gives the area. This involves determining the determinant of a matrix that includes the components of these vectors.

• Vector \(\textbf{A}\): \((a, b)\)
• Vector \(\textbf{B}\): \((c, d)\)

The absolute value of this determinant will give you the area of the parallelogram.

Determinant

A determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, such as: \[ \begin{vmatrix} a & b \ c & d \ \end{vmatrix} \] the determinant is calculated as: \[ \text{det} = ad - bc \] The absolute value of this determinant gives the area of the parallelogram formed by the vectors \(\textbf{A}\) and \(\textbf{B}\).
Determinants are crucial in linear algebra and play a significant role in various applications such as finding areas, volumes, and solving systems of linear equations. They help in understanding properties of matrices and transformations.

Vector Cross Product

The cross product of two vectors in 3D space results in a vector that is perpendicular to the plane containing the original vectors. However, for two-dimensional vectors, the cross product can be represented as a determinant. For vectors \(\textbf{A} = a \textbf{i} + b \textbf{j}\) and \(\textbf{B} = c \textbf{i} + d \textbf{j}\), the cross product is computed using the determinant of a matrix formed by their components: \[ \textbf{A} \times \textbf{B} = \begin{vmatrix} a & b \ c & d \ \end{vmatrix} = ad - bc \] The magnitude of this cross product gives the area of the parallelogram formed by the vectors.
Key points about the cross product:

• It is anti-commutative: \(\textbf{A} \times \textbf{B} = - (\textbf{B} \times \textbf{A})\)
• It is distributive: \( \textbf{A} \times (\textbf{B} + \textbf{C}) = (\textbf{A} \times \textbf{B}) + (\textbf{A} \times \textbf{C}) \)
• The result is a vector perpendicular to the plane of \(\textbf{A}\) and \(\textbf{B}\)

In the case of 2D vectors, it simplifies to the determinant calculation.

Matrix Representation

Matrices are organized sets of numbers arranged in rows and columns. In this exercise, we use a 2x2 matrix to represent the components of vectors forming a parallelogram. The matrix looks like this: \[ \begin{pmatrix} a & b \ c & d \ \end{pmatrix} \] Each element in the matrix corresponds to a component of the vectors. By finding the determinant of this matrix, we can determine geometric properties such as area.
Key properties of matrices:

• Matrices can represent transformations.
• They are used for solving systems of linear equations.
• Matrix operations include addition, multiplication, and finding the determinant.

In our case, the determinant of the above 2x2 matrix directly relates to the area of a parallelogram formed by the vectors with components represented in the matrix.