Problem 14 - 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. $$ \mathbf{u}=\mathbf{j}+2 \mathbf{k}, \quad \mathbf{v}=\mathbf{i} $$

Step-by-Step Solution

Answer

Vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{u} \times \mathbf{v}\) are plotted as \(\langle 0, 1, 2 \rangle\), \(\langle 1, 0, 0 \rangle\), and \(\langle 0, 2, -1 \rangle\).

Step 1: Understanding the Vectors

First, identify the components of the given vectors. The vector \(\mathbf{u} = \mathbf{j} + 2\mathbf{k}\) indicates that \(\mathbf{u}\) is in the form of \(\langle 0, 1, 2 \rangle\). Similarly, the vector \(\mathbf{v} = \mathbf{i}\) indicates that \(\mathbf{v}\) is in the form of \(\langle 1, 0, 0 \rangle\).

Step 2: Sketch the Coordinate Axes

Draw a 3D set of axes. Label the x-axis, y-axis, and z-axis. This will help you visualize the orientation and direction of the vectors.

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

Start at the origin \((0,0,0)\), move 0 along the x-axis, 1 unit along the y-axis, and 2 units along the z-axis. Draw the vector \(\mathbf{u}\) with an arrow indicating its direction.

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

Start at the origin \((0,0,0)\) and move 1 unit along the x-axis while remaining at 0 on the y-axis and z-axis. Draw the vector \(\mathbf{v}\) in this position with an arrow indicating its direction.

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

Use the cross product formula \(\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 0 & 1 & 2 \ 1 & 0 & 0 \end{vmatrix}\). Expanding the determinant: \(\mathbf{u} \times \mathbf{v} = \mathbf{i}(1 \cdot 0 - 2 \cdot 0) - \mathbf{j}(0 \cdot 0 - 2 \cdot 1) + \mathbf{k}(0 \cdot 0 - 1 \cdot 1) = 0\mathbf{i} + 2\mathbf{j} - 1\mathbf{k} \). The result is \(\langle 0, 2, -1 \rangle\).

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

Starting from the origin \((0,0,0)\), move 0 along the x-axis, 2 units along the y-axis, and -1 unit along the z-axis. Draw the cross product vector with an arrow to indicate its direction.

Key Concepts

Cross Product

The cross product is a fundamental operation in vector algebra. It allows us to find a new vector that is perpendicular to two given vectors in three-dimensional space. The cross product of two vectors, say \( \mathbf{u} \) and \( \mathbf{v} \), is denoted as \( \mathbf{u} \times \mathbf{v} \).
This operation is widely used in physics, computer graphics, and engineering to determine memorable properties like torque, angular momentum, or the normal to a surface.

• The cross product of two vectors results in a new vector that is orthogonal to both original vectors.
• The magnitude of the cross product vector represents the area of the parallelogram generated by the original vectors.

To calculate the cross product, we use the formula:
\[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 \end{vmatrix} \]
This third notion of vectors and their cross product is vital in understanding the relationship between different spatial directions.

Vector Components

Vectors are crucial to mathematics and physics as they describe quantities that possess both magnitude and direction. Each vector can be broken down into three components relative to 3D space axes, typically labeled x, y, and z.

Understanding vector components is essential for plotting and calculating with vectors effectively.

• The component along the x-axis is called the i-component, represented by \( \mathbf{i} \).
• The component along the y-axis is called the j-component, written as \( \mathbf{j} \).
• The component along the z-axis is the k-component, designated as \( \mathbf{k} \).

For example, a vector \( \mathbf{u} = 0\mathbf{i} + 1\mathbf{j} + 2\mathbf{k} \) implies that the vector has no displacement along the x-axis, a unit displacement along the y-axis, and two units along the z-axis. Understanding how to read and interpret these components is crucial for visualizing and utilizing vectors in problem-solving.

Plotting Vectors

Plotting vectors, especially in 3D, involves representing them accurately with respect to the coordinate axes. Visualizing vectors helps to better understand their directions and relative positions.

This can initially be challenging due to the added dimension compared to 2D plotting.

• Begin by drawing the x, y, and z axes in a way that shows depth, typically with the z-axis emerging out towards the viewer.
• Start each vector at the origin unless otherwise specified and follow the components in sequence.
• Move according to the values of the vector's components, laying them out step by step. For a vector \( \langle 1, 0, 0 \rangle \), move 1 unit along the x-axis, and there is no subsequent movement along the y or z.

When dealing with the cross product, the resultant vector is plotted to show its perpendicular nature. Remember, agility in plotting vectors grows with practice and can significantly aid in understanding complex spatial problems.