Problem 35 - Thomas Calculus

Question

Find the areas of the parallelograms whose vertices are given in Exercises \(35-40 .\) $$ A(1,0), \quad B(0,1), \quad C(-1,0), \quad D(0,-1) $$

Step-by-Step Solution

Answer

The area of the parallelogram is 2 square units.

Step 1: Understanding the Coordinates

The vertices of the parallelogram are given as \(A(1,0), B(0,1), C(-1,0), D(0,-1)\). We will use these points to determine vectors representing two adjacent sides of the parallelogram.

Step 2: Find Vectors for Adjacent Sides

To find vectors representing two adjacent sides of the parallelogram, we consider the vectors \( \vec{AB} \) and \( \vec{AD} \). Calculate these vectors:\[ \vec{AB} = B - A = (0 - 1, 1 - 0) = (-1, 1) \]\[ \vec{AD} = D - A = (0 - 1, -1 - 0) = (-1, -1) \]

Step 3: Calculate Cross Product Magnitude

The area of the parallelogram formed by the vectors \( \vec{AB} = (-1, 1) \) and \( \vec{AD} = (-1, -1) \) is given by the magnitude of the cross product of these vectors:For vectors \( \vec{u} = (x_1, y_1) \) and \( \vec{v} = (x_2, y_2) \), the cross product in 2D is \( |x_1y_2 - x_2y_1| \).Apply this formula:\[ |(-1)(-1) - (-1)(1)| = |1 + 1| = 2 \]

Step 4: Conclusion

The magnitude of the cross product is the area of the parallelogram. Therefore, the area of the parallelogram with vertices \( A(1,0), B(0,1), C(-1,0), D(0,-1) \) is 2 square units.

Key Concepts

Vectors

Vectors play a crucial role in understanding geometry, especially when it comes to calculating areas of shapes like parallelograms. A vector is essentially a mathematical object that has both magnitude (length) and direction.
They can be visualized as arrows pointing from one position to another in space. In two-dimensional space, a vector can be represented using coordinates, for instance, \((x, y)\), where \(x\) is the horizontal component and \(y\) is the vertical component.

• To define a vector from one point to another, simply subtract the coordinates of the starting point from the endpoint.
• For example, to find vector \(\vec{AB}\) from point \(A(1,0)\) to \(B(0,1)\), subtract \((1,0)\) from \((0,1)\) to get \((-1,1)\).

By using vectors, we can represent the sides of geometric figures, making it easier to calculate properties such as area.

Cross Product

The cross product is a mathematical operation used primarily to calculate the area of parallelograms in vector geometry. It involves taking two vectors and producing a third vector, which is perpendicular to the plane containing the first two.
However, when dealing with 2D vectors, what we usually compute is the magnitude of this imaginary third vector. This gives us the "area" concept for determining the magnitude of their failure to be aligned.

• For 2D vectors \(\vec{u} = (x_1, y_1)\) and \(\vec{v} = (x_2, y_2)\), the cross product magnitude is calculated as \(|x_1y_2 - x_2y_1|\).
• This formula can be thought of as a specific case of a determinant, providing the area of the parallelogram formed by the two vectors.

In this particular exercise, the vectors \(\vec{AB} = (-1, 1)\) and \(\vec{AD} = (-1, -1)\) intersect to form a specific angle, and the cross product tells us the area, yielding 2 square units.

Coordinates Geometry

Coordinate geometry, also known as analytic geometry, is a field of mathematics that uses algebraic equations to represent geometric figures. This discipline provides a link between algebra and geometry, allowing for a deeper understanding of spatial relationships.
In coordinate geometry, points, lines, and shapes can be represented on the Cartesian plane using coordinates, which simplifies the study of their properties.

• Points in coordinate geometry are defined by ordered pairs, such as \((x, y)\).
• Lines can be expressed by equations such as \(y = mx + b\), where \(m\) is the slope and \(b\) is the intercept point.
• Shapes like parallelograms are defined by their vertices' coordinates, allowing easy calculation of dimensions and areas.

The given problem uses coordinate geometry extensively, enabling the determination of vectors and calculation of the parallelogram's area using these representations. By leveraging coordinate geometry, complex geometrical problems are made much more accessible.