Problem 10 - 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. $$ \mathbf{u}=\mathbf{i}-\mathbf{k}, \quad \mathbf{v}=\mathbf{j} $$

Step-by-Step Solution

Answer

Plot vectors \(\mathbf{u} = (1,0,-1)\), \(\mathbf{v} = (0,1,0)\), and \(\mathbf{u} \times \mathbf{v} = (1,1,1)\) on the coordinate axes.

Step 1: Sketch the Coordinate Axes

Draw three-dimensional axes on paper, labeling them as the x-axis, y-axis, and z-axis. The intersection point where all axes meet is the origin (0,0,0).

Step 2: Plot Vector \(\mathbf{u}\)

Vector \(\mathbf{u} = \mathbf{i} - \mathbf{k}\) means 1 unit in the x-direction and -1 unit in the z-direction starting from the origin. Draw a line from the origin (0,0,0) to the point (1,0,-1).

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

Vector \(\mathbf{v} = \mathbf{j}\) represents 1 unit in the direction of the y-axis. Starting from the origin, draw a line to the point (0,1,0).

Step 4: Determine the Cross Product \(\mathbf{u} \times \mathbf{v}\)

Calculate the cross product of vectors \(\mathbf{u}\) and \(\mathbf{v}\):\[\mathbf{u} \times \mathbf{v} = (1, 0, -1) \times (0, 1, 0) = (1\cdot 0 + 0\cdot 0 - (0\cdot 0 - 1\cdot 1), 1\cdot 0 - (-1)\cdot 1, 1\cdot 1 - 0\cdot 0) = (1, 1, 1) = \mathbf{i} + \mathbf{j} + \mathbf{k}.\]

Step 5: Plot Vector \(\mathbf{u} \times \mathbf{v}\)

Now plot this resulting vector \(\mathbf{u} \times \mathbf{v} = \mathbf{i} + \mathbf{j} + \mathbf{k}\), which starts at the origin and points to (1,1,1).

Key Concepts

Three-Dimensional Vectors: Understanding the Basics

In the world of mathematics and physics, three-dimensional vectors play a crucial role in describing motion and positions in space. These vectors are represented using three components, each corresponding to one of the three dimensions: the x-axis, the y-axis, and the z-axis. \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \) is a typical format for a 3D vector, where \( a, b, \) and \( c \) are the magnitudes in the x, y, and z directions, respectively.
Understanding these components helps in visualizing and manipulating vectors in a spatial environment. For instance, in the exercise given, vector \( \mathbf{u} = \mathbf{i} - \mathbf{k} \) indicates the presence of 1 unit in the x-direction and -1 unit in the z-direction, while vector \( \mathbf{v} = \mathbf{j} \) is simply 1 unit in the y-direction. This comprehension allows us to visualize how objects or entities move in a three-dimensional space.

Coordinate Axes: The Framework of 3D Space

The coordinate axes are fundamental to understanding and visualizing three-dimensional vectors. Typically, these axes consist of the x-axis, y-axis, and z-axis. They intersect at a point called the origin, denoted as (0, 0, 0).

• The x-axis is often drawn horizontally and represents the left-to-right dimension.
• The y-axis is vertical, indicating the upward or downward dimension.
• The z-axis is usually depicted coming out of or going into the plane, representing the in-and-out dimension.

Familiarity with these axes allows us to plot vectors accurately. For example, plotting \( \mathbf{u} = \mathbf{i} - \mathbf{k} \) involves moving 1 unit along the x-axis and -1 unit along the z-axis. This foundational knowledge of the coordinate axes builds a framework upon which complex calculations and visualizations can be understood.

Vector Visualization: Bringing Math to Life

Visualizing vectors helps in interpreting mathematical expressions in the context of spatial environments. Using the coordinate axes, vectors can be drawn as arrows starting from the origin to a specific point, hence making interpretation more intuitive.
When given vectors \( \mathbf{u} \) and \( \mathbf{v} \) as in the exercise, imagining them in space becomes crucial. Each vector points in a specific direction according to its components. For instance, plotting \( \mathbf{u} = \mathbf{i} - \mathbf{k} \) on the coordinate axes shows it going towards the point (1, 0, -1). The cross product \( \mathbf{u} \times \mathbf{v} = \mathbf{i} + \mathbf{j} + \mathbf{k} \) results in a new vector that points to (1, 1, 1), illustrating how interaction between vectors produces a vector orthogonal to both.
This visualization process not only aids in understanding vector properties but also demonstrates their practical applications in various real-world scenarios.