Problem 12 - 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. $$\mathbf{u}=2 \mathbf{i}-\mathbf{j}, \quad \mathbf{v}=\mathbf{i}+2 \mathbf{j}$$

Step-by-Step Solution

Answer

The vectors \( \mathbf{u} \) and \( \mathbf{v} \) are plotted on the XY plane, while \( \mathbf{u} \times \mathbf{v} \) points out of the page in the positive Z-direction.

Step 1: Sketching the Coordinate Axes

First, draw a traditional 2D XY coordinate plane. The horizontal line is the X-axis, and the vertical line is the Y-axis. These axes will serve as your reference for plotting vectors.

Step 2: Plotting the Vector \( \mathbf{u} \)

The vector \( \mathbf{u} = 2 \mathbf{i} - \mathbf{j} \) can be written as \( (2, -1) \). Starting from the origin (0,0), move 2 units right (along the X-axis) and 1 unit down (along the Y-axis) to reach the point (2, -1). Draw an arrow from the origin to this point to represent \( \mathbf{u} \).

Step 3: Plotting the Vector \( \mathbf{v} \)

The vector \( \mathbf{v} = \mathbf{i} + 2 \mathbf{j} \) can be expressed as \( (1, 2) \). Starting from the origin (0,0), move 1 unit right on the X-axis and 2 units up on the Y-axis to reach point (1, 2). Draw an arrow from the origin to this point to represent \( \mathbf{v} \).

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

The cross product of two vectors in 3D is only meaningful. Extend the 2D vectors to 3D by adding a 0-component in the z-direction: \( \mathbf{u} = (2, -1, 0) \) and \( \mathbf{v} = (1, 2, 0) \). Using the right-hand rule, calculate \( \mathbf{u} \times \mathbf{v} \) as a determinant:\[\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \2 & -1 & 0 \1 & 2 & 0 \end{vmatrix}\]This results in:\[\mathbf{u} \times \mathbf{v} = 0\mathbf{i} - 0\mathbf{j} + 5\mathbf{k} = 5\mathbf{k}\].

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

The vector \( \mathbf{u} \times \mathbf{v} = 5\mathbf{k} \) points in the positive Z-direction in 3D space. On the 2D coordinate plane, indicate this as an arrow maintaining the origin (0,0) as its start and pointing outwards (perpendicular to the page) to reflect the third dimension.

Key Concepts

Understanding Coordinate Axes

Coordinate axes are fundamental to visualizing vectors in space. Think of them as a map grid, helping you know exactly where you're placing points and lines. In a 2D plane, the coordinate axes consist of the horizontal X-axis and the vertical Y-axis. These axes intersect at a point called the origin, typically labeled as (0,0). In this exercise, these axes provide a reference frame to help us plot our vectors.

• **X-axis**: This axis runs horizontally. Positive directions move to the right.
• **Y-axis**: This axis runs vertically. Positive directions move upwards.
• **Origin**: The point where both axes meet, labeled as (0, 0).

Whenever solving a problem involving vectors, drawing these axes helps a lot! By using these guidelines, you can plot vectors starting from the origin efficiently.

Exploring the Vector Cross Product

The vector cross product is a fascinating operation that results in a vector perpendicular to the plane containing the two original vectors. It's particularly important in 3D space, where understanding such relationships can be visualized through physics or engineering applications.The cross product of vectors \( \mathbf{u} \) and \( \mathbf{v} \) is computed using a matrix determinant, often resulting in a new vector that we can call \( \mathbf{u} \times \mathbf{v} \). This operation is seen through the formula:\[\mathbf{u} \times \mathbf{v} = (u_y v_z - u_z v_y)\mathbf{i} - (u_x v_z - u_z v_x)\mathbf{j} + (u_x v_y - u_y v_x)\mathbf{k}\]This formula might seem a bit complex at first. However, it effectively rotates vectors into a new space, often depicted in advanced 3D representations, beyond basic 2D vectors. Through the cross product in this exercise, the vector \(\mathbf{u} \times \mathbf{v}\) points in the Z direction, embodying the perpendicularity principle.

3D Vector Representation

Transitioning from a 2D to a 3D representation opens a whole new realm for vectors. Here, we introduce a third axis: the Z-axis, which extends perpendicular to both X and Y axes. Representing vectors in 3D involves adding the Z-component, written as \( (x, y, z) \).

• **Z-axis**: Completes the trio of axes, forming a three-dimensional grid.
• **3D Vector Notation**: Represents vectors with an additional third component for height/depth, making it \( (x, y, z) \).

In the provided exercise, this idea is vital when calculating the cross product because turning 2D vectors into 3D by adding \( z = 0 \) ensures that they can meaningfully interact in this extended space. The result, \( 5\mathbf{k} \), precisely follows this expansion, underlining the power of 3D representation.

Sketching Vectors

Visualizing vectors through sketching is an essential skill in understanding their properties and relationships. Sketching involves representing vectors as arrows with specific directions and magnitudes on a coordinate grid.

• **Origin Point**: Always start your vectors from the origin to maintain consistency.
• **Direction**: Determine the movement along axes (e.g., right, up, etc.).
• **Magnitude**: The length of the vector's arrow reflects its size.

Using visual tools can help make abstract concepts more tangible. When sketching vectors like \( \mathbf{u} = 2\mathbf{i} - \mathbf{j} \) and \( \mathbf{v} = \mathbf{i} + 2\mathbf{j} \), begin at the origin, correctly follow the directions given by each component, and mark endpoints clearly. This visual representation aids in easier comprehension of vector addition, subtraction, and the results of operations like the cross product.