Problem 27 - Essential Calculus Early Transcendentals

Question

Find the area of the parallelogram with vertices \(A(-2,1),\) \(B(0,4), C(4,2),\) and \(D(2,-1)\) .

Step-by-Step Solution

Answer

The area of the parallelogram is 16 square units.

Step 1: Understand the Formula for the Area

The area of the parallelogram formed by vectors \( \vec{AB} \) and \( \vec{AD} \) can be calculated using the magnitude of their cross product. The formula is given by: \[ \text{Area} = \left| \vec{AB} \times \vec{AD} \right| \] where \( \vec{AB} = (B_x - A_x, B_y - A_y) \) and \( \vec{AD} = (D_x - A_x, D_y - A_y) \).

Step 2: Find the Vectors \( \vec{AB} \) and \( \vec{AD} \)

Calculate \( \vec{AB} \) and \( \vec{AD} \):\[ \vec{AB} = (0 - (-2), 4 - 1) = (2, 3) \]\[ \vec{AD} = (2 - (-2), -1 - 1) = (4, -2) \]

Step 3: Compute the Cross Product of Vectors

The cross product of two vectors \((x_1, y_1)\) and \((x_2, y_2)\) in 2D is given by:\[ \vec{AB} \times \vec{AD} = x_1y_2 - x_2y_1 \]Substitute the values of \( \vec{AB} \) and \( \vec{AD} \):\[ \vec{AB} \times \vec{AD} = (2)(-2) - (4)(3) = -4 - 12 = -16 \]

Step 4: Calculate the Magnitude of the Cross Product

The area of the parallelogram is the absolute value of the cross product:\[ \text{Area} = |\vec{AB} \times \vec{AD}| = |-16| = 16 \]

Key Concepts

Understanding the Cross Product

The cross product is a mathematical operation that can be used to find the area of a parallelogram defined by two vectors. When dealing with vectors in a two-dimensional space, the cross product is calculated by taking the difference between the product of the opposite components of each vector. This operation results in a scalar, not a vector, which represents the magnitude or size that is perpendicular to the plane formed by these two vectors.
\[ \text{If} \; \vec{u} = (x_1, y_1) \; \text{and} \; \vec{v} = (x_2, y_2), \; \text{then} \; \vec{u} \times \vec{v} = x_1y_2 - x_2y_1 \]
The result, although a scalar for 2D, exactly provides us the magnitude necessary to compute the area of the parallelogram.

• **2D Cross Product:** Calculates a scalar
• **Represents Area:** Helps find the magnitude of a parallelogram
• **Orientation:** Directionality in 3D gives vector; in 2D, only magnitude

By understanding the cross product, students learn how direction and magnitude combine to determine geometrical properties like area.

Vectors: The Building Blocks

Vectors are essential in mathematics and physics as they describe a quantity possessing both magnitude and direction. They are represented graphically as arrows pointing from one point to another in space. Each vector has components along the x, y (and sometimes z) axes that are crucial in calculations.
For example, the vector \( \vec{AB} \) formed between points \(A(-2,1)\) and \(B(0,4)\) has components calculated as \((B_x - A_x, B_y - A_y)\), resulting in \(\vec{AB} = (2, 3)\). The same method applies to find \( \vec{AD} = (4, -2)\) from points \(A\) and \(D(2,-1)\).

• **Magnitude and Direction:** Core attributes of vectors
• **Component Calculation:** \((x, y)\) format for 2D space
• **Application:** Form the edges of geometrical shapes like parallelograms

In geometry, vectors help define shapes and spaces by creating representations of lines and edges, essential in calculating areas like that of a parallelogram.

Geometry: Parallelogram Area

Geometry revolves around shapes, sizes, and the properties of space. A parallelogram is a quadrilateral with two pairs of parallel sides, and finding its area is a frequent problem. This is where vectors and cross products combine because the cross product essentially quantifies the area of the parallelogram defined by two side vectors.
The area calculation follows from:
\[ \text{Area} = \left| \vec{AB} \times \vec{AD} \right| \]
This formula is derived from the properties of vectors and their cross product. As a result, we need to find vectors at the vertices of the parallelogram and use the cross product formula to compute the desired area.

• **Parallel Sides:** Characteristic feature of parallelograms
• **Area Calculation:** Utilize vectors and cross product
• **Simplified Approach:** Through algebra and vector geometry

By integrating concepts such as vectors and cross products, geometry offers more than a simple formula; it showcases a method to analyze spatial figures and their inherent properties.