Problem 13 - University Calculus: Early Transcendentals

Question

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

Sketch vectors \( \mathbf{u} \) and \( \mathbf{v} \) on xy-plane, and \( \mathbf{u} \times \mathbf{v} = 2\mathbf{k} \) along z-axis.

Step 1: Sketch the Coordinate Axes

Begin by drawing the coordinate axes, a two-dimensional plane where the horizontal line is labeled the x-axis and the vertical line is labeled the y-axis.

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

Vector \( \mathbf{u} \) is represented as \( \langle 1, 1 \rangle \), which means it moves 1 unit along the x-axis and 1 unit along the y-axis starting from the origin (0,0). Mark this point and draw an arrow from the origin to this point.

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

Vector \( \mathbf{v} \) is \( \langle 1, -1 \rangle \), indicating it moves 1 unit along the x-axis and -1 unit along the y-axis from the origin. Locate this point and draw an arrow from the origin to the new point.

Step 4: Calculate \( \mathbf{u} \times \mathbf{v} \)

The cross product \( \mathbf{u} \times \mathbf{v} \) in 2D is performed in a 3-space extension by assuming \( z \) components are 0: \( \mathbf{u} = \langle 1, 1, 0 \rangle \) and \( \mathbf{v} = \langle 1, -1, 0 \rangle \). Use the determinant formula for cross product: \[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 1 & 0 \ 1 & -1 & 0 \end{vmatrix} = \mathbf{k}(1-(-1)) = 2\mathbf{k} \].

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

Since \( \mathbf{u} \times \mathbf{v} = 2\mathbf{k} \), this vector is along the z-axis. Draw an arrow starting from the origin pointing upwards on the z-axis.

Key Concepts

Cross Product

A cross product is a method used in vector calculus to find a vector that is perpendicular to two input vectors. This process is essential in physics and engineering, especially when dealing with rotational forces and determining the area of a parallelogram formed by these vectors. When computing the cross product of two 2D vectors, such as \( \mathbf{u} \) and \( \mathbf{v} \), we extend them into 3-space by adding a zero \( z \)-component. For example, our vectors \( \mathbf{u} = \mathbf{i} + \mathbf{j} = \langle 1, 1, 0 \rangle \) and \( \mathbf{v} = \mathbf{i} - \mathbf{j} = \langle 1, -1, 0 \rangle \) become three-dimensional. To find the cross product \( \mathbf{u} \times \mathbf{v} \), we use the determinant formula:\[\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 1 & 0 \ 1 & -1 & 0 \end{vmatrix} \]This results in the vector \( 2\mathbf{k} \), perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \), lying on the \( z \)-axis.

• The cross product has important properties such as being:
  * Anticommutative: \( \mathbf{u} \times \mathbf{v} = - (\mathbf{v} \times \mathbf{u}) \)
  * Distributive over vector addition.
• Cross products produce vectors, unlike dot products which yield scalars.

Coordinate Systems

In mathematics and physics, coordinate systems are essential frameworks that allow us to define the position of a point in space. A simple 2D coordinate system is composed of two perpendicular axes:

• The \( x \)-axis, which is horizontal, commonly represents the set of real numbers.
• The \( y \)-axis, vertical, also represents the real numbers.

Together, these axes form a plane, and a point on this plane can be identified using a pair of coordinates \( (x, y) \). Starting from the origin, where the axes intersect at \( (0, 0) \), each point corresponds to a unique set of coordinates based on its distance from the axes. For vectors \( \mathbf{u} \) and \( \mathbf{v} \), we place them in this 2D coordinate system. They start at the origin, and their terminal points are based on their respective directional coordinates. The implementation of this system makes vector analysis calculations straightforward, allowing for a clear depiction of the vectors' positions and their relational geometry in space.

Vector Representation

Vectors are fundamental elements in mathematics, described in terms of direction and magnitude, being critical for understanding how quantities with both properties behave. In Cartesian coordinates, they are represented as a set of numbers which correspond to their direction along each axis. To represent a vector in a coordinate system, we use an origin point, typically \( (0, 0) \) in 2D, from which the vector is projected.For instance, two-dimensional vectors like \( \mathbf{u} = \mathbf{i} + \mathbf{j} = \langle 1, 1 \rangle \) indicate:

• The vector moves 1 unit on the \( x \)-axis (\( \mathbf{i} \)).
• It also moves 1 unit on the \( y \)-axis (\( \mathbf{j} \)).

Similarly, the vector \( \mathbf{v} = \mathbf{i} - \mathbf{j} = \langle 1, -1 \rangle \) moves:

• 1 unit on the \( x \)-axis (\( \mathbf{i} \)).
• -1 unit on the \( y \)-axis (\( \mathbf{j} \)), changing direction downward.

Vectors can be extended to 3D with the addition of a \( z \)-component, allowing even richer representation and resulting in application to a wider range of problems, especially in physics and engineering.