Problem 13 - 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. $$ \mathbf{u}=\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=\mathbf{i}-\mathbf{j} $$

Step-by-Step Solution

Answer

The vectors \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{u} \times \mathbf{v} \) are \((1, 1, 0)\), \((1, -1, 0)\), and \((0, 0, -2)\) respectively.

Step 1: Understand the Vectors

First, identify the given vectors \( \mathbf{u} \) and \( \mathbf{v} \). The vector \( \mathbf{u} \) is given by \( \mathbf{i} + \mathbf{j} \) which represents the vector \( (1, 1, 0) \) in 3D space. The vector \( \mathbf{v} \) is \( \mathbf{i} - \mathbf{j} \), which represents the vector \( (1, -1, 0) \) in 3D space.

Step 2: Draw the Coordinate Axes

Sketch the standard three-dimensional coordinate axes: the x-axis, y-axis, and z-axis. Typically, the x-axis is horizontal, the y-axis is vertical, and the z-axis points outwards from the page.

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

Plot the vector \( \mathbf{u} = (1, 1, 0) \) starting from the origin. This means you move 1 unit along the x-axis and 1 unit along the y-axis. There is no movement along the z-axis.

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

Plot the vector \( \mathbf{v} = (1, -1, 0) \) starting from the origin. Move 1 unit along the x-axis and \(-1\) unit along the y-axis. Like \( \mathbf{u} \), there is no movement along the z-axis.

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

Calculate the cross product \( \mathbf{u} \times \mathbf{v} \) using the determinant method: \[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 1 & 0 \ 1 & -1 & 0 \end{vmatrix} = (0)\mathbf{i} - (0) \mathbf{j} + (-2)\mathbf{k} = -2\mathbf{k}. \]Thus, \( \mathbf{u} \times \mathbf{v} = (0, 0, -2). \)

Step 6: Plot the Cross Product Vector

Plot the vector \( \mathbf{u} \times \mathbf{v} = (0, 0, -2) \) starting from the origin. This vector does not move along the x or y axes, but moves \(-2\) units along the z-axis, which extends downward.

Key Concepts

3D Coordinate System

The 3D coordinate system is an extension of the familiar 2D coordinate system, adding another dimension to represent objects in space. In this system, we have three axes: the x-axis, the y-axis, and the z-axis.
Each axis is perpendicular to the other two, forming a three-dimensional space where any point can be located using three coordinates (x, y, z).

• x-axis: Typically represented as running horizontally.
• y-axis: Usually the vertical axis.
• z-axis: Perpendicular to both x and y axes, often visualized as coming out of or going into the page/screen.

In a 3D space, vectors can have components along each of these axes. For instance, a vector (1, 1, 0) indicates a movement of 1 unit along the x-axis, 1 unit along the y-axis, and no movement along the z-axis. This system allows us to visualize complex geometric ideas like lines, planes, and rotations.

Vector Plotting

Vector plotting in a 3D coordinate system involves drawing vectors as arrows from a starting point, commonly the origin (0,0,0). Each vector is defined by its components, which tell us how much the vector travels along each axis.
For example, the vector \( \mathbf{u} = (1, 1, 0) \) starts at the origin and moves 1 unit right (along the x-axis) and 1 unit up (along the y-axis), without any movement into or out of the page (along the z-axis).

• Starting Point: Typically, vectors are plotted starting at the origin, but they can start at any point, maintaining the same direction and length.
• Direction and Length: The direction is given by the line the vector follows, and its length is the magnitude, calculated as \( \sqrt{x^2 + y^2 + z^2} \).

When multiple vectors are involved, plotting each vector can help visualize operations like addition, subtraction, or the cross product. Seeing these in 3D helps in understanding their interactions and geometric interpretations.

Vector Algebra

Vector algebra is a branch of mathematics dealing with operations on vectors, such as addition, subtraction, and multiplication. In the context of vector algebra, we use vectors to represent physical quantities like force or velocity, among others.
When working with vectors, each operation has a specific geometric significance:

• Addition: Two vectors are added by joining them head to tail, forming a new vector from the start of the first to the end of the second.
• Subtraction: Achieved by adding the negative of a vector, which is a vector of the same magnitude but opposite direction.
• Scalar Multiplication: Involves multiplying a vector by a scalar, affecting only its magnitude, not its direction.

Vector multiplication can be of two types: dot product and cross product. The cross product, in particular, is relevant in 3D operations and results in a vector perpendicular to the two multiplied vectors, which is especially useful in physics and engineering.

Determinant Method

The determinant method is a mathematical process used to calculate the cross product of two vectors in 3D space. The cross product itself is a vector that is perpendicular to both of the initial vectors, and its direction is given by the right-hand rule.
To find the cross product of two vectors, say \( \mathbf{u} \) and \( \mathbf{v} \), we set up a 3x3 determinant:
\[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ u_x & u_y & u_z \ v_x & v_y & v_z \end{vmatrix} \]
Here, \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) are the unit vectors in the direction of the x, y, and z axes, respectively. The components of \( \mathbf{u} \) and \( \mathbf{v} \) fill the matrix.
The determinant expands to a new vector found by:

• Using the minors of each component of the first row (\( \mathbf{i}, \mathbf{j}, \mathbf{k} \)).
• Alternating addition and subtraction for each minor.
• The result is a vector \( (A, B, C) \), where each component is a combination of the components of \( \mathbf{u} \) and \( \mathbf{v} \).

This method is pivotal as it unifies vector algebra with linear algebra techniques, providing a geometric understanding of vector operations.