Problem 12 - Thomas Calculus

Question

In Exercises \(9-14,\) sketch the coordinate axes and then include the vectors \(\mathbf{u}, \mathbf{v}\) and \(\mathbf{u} \times \mathbf{v}\) as vectors starting at the origin. $$ \mathbf{u}=2 \mathbf{i}-\mathbf{j}, \quad \mathbf{v}=\mathbf{i}+2 \mathbf{j} $$

Step-by-Step Solution

Answer

Draw vectors \( \mathbf{u} \) and \( \mathbf{v} \) at the origin on a 2D plane. The cross product results in a perpendicular scalar.

Step 1: Understand Vector Representation

Vectors are represented in terms of the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \). For vector \( \mathbf{u} = 2 \mathbf{i} - \mathbf{j} \), it translates to moving 2 units along the x-axis and -1 unit along the y-axis. Similarly, vector \( \mathbf{v} = \mathbf{i} + 2 \mathbf{j} \) moves 1 unit along the x-axis and 2 units along the y-axis.

Step 2: Draw Coordinate Axes

Sketch the x-axis and y-axis on a graph. These axes will help us plot the vectors starting from the origin (0,0).

Step 3: Plot Vector \( \mathbf{u} \)

Starting from the origin, draw the vector \( \mathbf{u} = 2 \mathbf{i} - \mathbf{j} \). Move 2 units to the right on the x-axis and then move 1 unit down on the y-axis. Mark this endpoint and draw an arrow from the origin to this point.

Step 4: Plot Vector \( \mathbf{v} \)

From the origin, draw the vector \( \mathbf{v} = \mathbf{i} + 2 \mathbf{j} \). Move 1 unit to the right on the x-axis and 2 units up on the y-axis. Draw an arrow from the origin to this new point.

Step 5: Calculate Cross Product \( \mathbf{u} \times \mathbf{v} \)

The cross product \( \mathbf{u} \times \mathbf{v} \) in 2D space typically results in a scalar, indicating direction perpendicular to the plane of \( \mathbf{u} \) and \( \mathbf{v} \). Thus, ignore for drawing purposes as the representation lies in a separate three-dimensional axis.

Key Concepts

Coordinate System

A coordinate system is essential for understanding the positions and movements of vectors. It consists of two perpendicular lines, usually labeled as the x-axis and y-axis, intersecting at a point called the origin. Vectors are plotted using this system, allowing us to express their direction and magnitude within a defined framework.

• The x-axis typically represents horizontal movement.
• The y-axis represents vertical movement.
• Each point on the plane can be identified with coordinate pairs \(x, y\).

In textbooks and other exercises, coordinate systems are vital for visualizing how vectors interact and overlap within a specific space. For example, when drawing vectors \(\mathbf{u}\) and \(\mathbf{v}\) from an origin point, the coordinate system helps determine their paths, and each movement along the axes is clear and concise.

Cross Product

The cross product is a fundamental operation for vectors in three-dimensional space. For the vectors given in our exercise, \(\mathbf{u} = 2 \mathbf{i} - \mathbf{j}\) and \(\mathbf{v} = \mathbf{i} + 2 \mathbf{j}\), this operation usually produces a result perpendicular to both vectors involved. In simpler terms, think of the cross product as finding a vector that "points outwards" from the plane formed by the initial vectors, \(\mathbf{u}\) and \(\mathbf{v}\), rather like identifying the flagpole holding two flags fluttering in different directions. Here's how it works:

• Produces a vector perpendicular to both original vectors.
• Calculated using \(\mathbf{u} \times \mathbf{v}\), this often results in a scalar that points in the direction normal to the plane of the vectors.
• In 2D problems, the true cross product "result vector" happens in the third dimension, which is not part of the original visualization.

It's important to note that when working purely in a 2D space, like in this exercise, the cross product results in a scalar and isn't visually plottable using the original graph used for \(\mathbf{u}\) and \(\mathbf{v}\).

Unit Vectors

Unit vectors are the building blocks of vector expression in mathematics. They have a magnitude of one and serve as direction indicators within the coordinate plane. In the given exercise, the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are expressed using unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). Here's why unit vectors are indispensable:

• Standard Direction Indication: They provide a consistent way to specify directions along the coordinate axes.
• Simplicity of Calculations: Vectors can be easily calculated when expressed in terms of \(\mathbf{i}\) and \(\mathbf{j}\).
• Scalability: Any vector can be broken down into components parallel to the unit vectors, aiding in visual and calculation tasks.

For instance, unit vector \(\mathbf{i}\) points along the x-axis, while \(\mathbf{j}\) points along the y-axis. This makes calculating different vector components and determining their resultant directions simpler. When you express \(\mathbf{u}\) as \(2 \mathbf{i} - \mathbf{j}\), you know precisely where it moves on the graph, thanks to these unit vectors.