Problem 11 - 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. $$\mathbf{u}=\mathbf{i}-\mathbf{k}, \quad \mathbf{v}=\mathbf{j}+\mathbf{k}$$

Step-by-Step Solution

Answer

Plot vectors \(\mathbf{u} = \mathbf{i} - \mathbf{k}\), \(\mathbf{v} = \mathbf{j} + \mathbf{k}\), and their cross product \(\mathbf{i} - \mathbf{j} + \mathbf{k}\) on the coordinate axes starting from the origin.

Step 1: Identify the Components of the Vectors

Vector \(\mathbf{u} = \mathbf{i} - \mathbf{k}\) which indicates \(x = 1\), \(y = 0\), and \(z = -1\). Vector \(\mathbf{v} = \mathbf{j} + \mathbf{k}\) which shows \(x = 0\), \(y = 1\), and \(z = 1\). These components will help us to plot the vectors.

Step 2: Sketch the Coordinate Axes

Draw three perpendicular axes on a graph paper or chart, labeling them \(x\), \(y\), and \(z\) respectively. These will act as the reference frame for plotting the vectors.

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

Starting at the origin (0,0,0), plot the vector \(\mathbf{u} = \mathbf{i} - \mathbf{k}\). Move 1 unit along the \(x\)-axis and then -1 unit along the \(z\)-axis. Connect this point back to the origin with a directed line segment.

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

From the origin, plot the vector \(\mathbf{v} = \mathbf{j} + \mathbf{k}\). Move 1 unit along the \(y\)-axis and 1 unit along the \(z\)-axis. Draw a directed line segment from the origin to this point.

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

Use the cross product formula: \[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 0 & -1 \ 0 & 1 & 1 \end{vmatrix} = \mathbf{i}(0\cdot1 - (-1)\cdot1) - \mathbf{j}(1\cdot1 - (-1)\cdot0) + \mathbf{k}(1\cdot1 - 0\cdot0) \]Simplifying, this becomes: \(\mathbf{u} \times \mathbf{v} = \mathbf{i}(1) - \mathbf{j}(1) + \mathbf{k}(1) = \mathbf{i} - \mathbf{j} + \mathbf{k}\).

Step 6: Plot Vector \(\mathbf{u} \times \mathbf{v}\)

Starting at the origin, plot the vector \(\mathbf{u} \times \mathbf{v} = \mathbf{i} - \mathbf{j} + \mathbf{k}\). Move 1 unit along the \(x\)-axis, -1 unit along the \(y\)-axis, and 1 unit along the \(z\)-axis. Draw a directed line segment from the origin to this point.

Step 7: Review and Label the Vectors

Check that all vectors are correctly plotted starting from the origin and clearly label each vector \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{u} \times \mathbf{v}\) on the sketch.

Key Concepts

Vector Components

Vectors are essential elements in vector calculus, representing both a magnitude and direction. A vector's components break it down into its influence along the standard basis unit vectors: \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\), which align with the \(x\), \(y\), and \(z\) axes respectively. This decomposition allows us to easily understand and manipulate vectors in three-dimensional space.

For example, the vector \(\mathbf{u} = \mathbf{i} - \mathbf{k}\) indicates that it has components of 1 along the \(x\)-axis, 0 along the \(y\)-axis, and -1 along the \(z\)-axis. Similarly, \(\mathbf{v} = \mathbf{j} + \mathbf{k}\) has 0 along the \(x\)-axis, 1 along the \(y\)-axis, and 1 along the \(z\)-axis.

This breakdown:

• Helps visualize and plot vectors easily.
• Enables operations like addition, subtraction, and cross products.
• Facilitates the understanding of their physical implication, such as velocity along different directions in physics.

Coordinate Axes

The coordinate axes are fundamental to drawing and understanding vectors in three-dimensional space. They provide a reference frame that allows each vector component to be expressed in terms of its influence along perpendicular directions.

These axes are defined as:

• The \(x\)-axis, associated with unit vector \(\mathbf{i}\), and represents movement left or right.
• The \(y\)-axis, corresponding to unit vector \(\mathbf{j}\), and denotes vertical movement.
• The \(z\)-axis, aligned with unit vector \(\mathbf{k}\), which directs into or out of the drawing surface.

When sketching, it's important to draw these axes at right angles to one another, which sets up the Cartesian coordinate system. This system allows you to visualize vectors, like \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{u} \times \mathbf{v}\), in relation to one another clearly.

Cross Product

The cross product is a vector operation that produces a vector perpendicular to two given vectors in a three-dimensional space. It’s a useful tool for finding normal vectors to surfaces or for calculating torques in physics.

Mathematically, the cross product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is given by the determinant of a matrix composed of the standard unit vectors and the components of the vectors:\[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 \end{vmatrix} \]
This results in:

• A vector whose direction is determined by the right-hand rule: pointing your thumb in the direction of \(\mathbf{u}\) and your fingers in the direction of \(\mathbf{v}\), your palm faces the direction of \(\mathbf{u} \times \mathbf{v}\).
• A vector whose magnitude is given by \(\|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\|\|\mathbf{v}\|\sin \theta\), where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\).

In the exercise, the resulting vector \(\mathbf{u} \times \mathbf{v} = \mathbf{i} - \mathbf{j} + \mathbf{k}\) shows this principle, being orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\). This showcases how cross products encapsulate not only an operation, but yield meaningful geometric information as well.