Problem 23 - APEX Calculus

Question

In Exercises 23-26, find the area of the parallelogram defined by the given vectors. \(\vec{u}=\langle 1,1,2\rangle, \quad \vec{v}=\langle 2,0,3\rangle\)

Step-by-Step Solution

Answer

The area of the parallelogram is \(\sqrt{14}\).

Step 1: Understand the Problem

The problem is asking us to find the area of a parallelogram defined by two vectors, \(\vec{u}\) and \(\vec{v}\), in three dimensions. The formula for the area of a parallelogram with vectors \(\vec{u}\) and \(\vec{v}\) is the magnitude of their cross product \(\vec{u} \times \vec{v}\).

Step 2: Compute the Cross Product

The cross product of two vectors \(\vec{u} = \langle 1,1,2 \rangle\) and \(\vec{v} = \langle 2,0,3 \rangle\) can be found using the determinant:\[\vec{u} \times \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 1 & 1 & 2 \ 2 & 0 & 3 \end{vmatrix} = \hat{i}(1 \cdot 3 - 2 \cdot 0) - \hat{j}(1 \cdot 3 - 2 \cdot 2) + \hat{k}(1 \cdot 0 - 1 \cdot 2)\]Simplifying each term, we get:\[3\hat{i} - (-1)\hat{j} - 2\hat{k} = \langle 3, 1, -2 \rangle\]

Step 3: Calculate the Magnitude of the Cross Product

The magnitude of the vector \(\langle 3, 1, -2 \rangle\) is the area of the parallelogram, calculated as:\[\| \vec{u} \times \vec{v} \| = \sqrt{3^2 + 1^2 + (-2)^2} = \sqrt{9 + 1 + 4} = \sqrt{14}\]

Step 4: Interpret the Result

The magnitude you computed represents the area of the parallelogram defined by vectors \(\vec{u}\) and \(\vec{v}\). Therefore, the area of the parallelogram is \(\sqrt{14}\).

Key Concepts

Vector Cross Product

The vector cross product is an operation that takes in two vectors and produces another vector that is perpendicular to both. This mathematical tool is vital when working with three-dimensional spaces. The result of a cross product is a vector whose direction is given by the right-hand rule. This means if you point your index finger in the direction of the first vector, and your middle finger in the direction of the second vector, your thumb points in the direction of the cross product.

• The cross product is widely used to calculate areas of parallelograms when vectors are in play.
• It retains the properties of being non-commutative, meaning \(\vec{u} \times \vec{v} eq \vec{v} \times \vec{u} \).
• Mathematically, it’s calculated using a determinant involving the unit vectors \(\hat{i}, \hat{j}, \hat{k}\) in three-dimensional space.

Magnitude of a Vector

To understand the "magnitude of a vector," think of it as the length or size of a vector. In practice, it tells you how long or big a vector is, regardless of its direction. You can think of this like measuring the distance from the start of a vector to its tip.

• The magnitude is found by taking the square root of the sum of the squares of its components.
• For a vector \(\langle a, b, c \rangle\), the magnitude is denoted as \(\| \vec{v} \|\) and is calculated as \(\sqrt{a^2 + b^2 + c^2}\).

When you calculate the cross product of two vectors, taking the magnitude of the resulting vector gives you important information. Specifically, it tells you the area of the parallelogram formed by the original vectors.

Determinant in Vector Calculus

The determinant is a critical mathematical concept that helps solve several problems in vector calculus and linear algebra. When it comes to vectors, it's especially useful in computing the cross product.
To find the cross product using the determinant, think of a 3x3 matrix:

• The first row contains unit vectors \(\hat{i}, \hat{j}, \hat{k}\).
• The second row has the components of the first vector.
• The third row includes components of the second vector.

By evaluating this determinant, you can obtain the components of the cross product vector. The determinant gives a clear-cut formula to transform a complex vector process into manageable calculations. It ensures that the resulting vector is orthogonal (perpendicular) to the original vectors.

Three-Dimensional Vectors

Three-dimensional vectors describe quantities that have both magnitude and direction in a space defined by three axes: x, y, and z. These are abundant in physics and engineering, representing entities like force, velocity, and movement.

• Each vector can be expressed in the form \(\langle x, y, z \rangle\) where \(x, y, z\) are the vector's components along the respective axes.
• These vectors are needed when solving problems involving angles, areas, and distances in three-dimensional space.

They become especially significant when calculating the cross product, since this process is only defined in three or more dimensions. Understanding how vectors operate in this space lays the foundation for working with more sophisticated vector calculations.