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

Draw \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{u} \times \mathbf{v} \) on the coordinate axes based on their components.

Step 1: Understand Vector Components

The vector \( \mathbf{u} = \mathbf{j} + 2\mathbf{k} \) means that it has no component in the x-direction, 1 component in the y-direction, and 2 components in the z-direction. The vector \( \mathbf{v} = \mathbf{i} \) means that it has 1 component in the x-direction, and no components in the y- or z-directions.

Step 2: Draw Vector u

On the 3D coordinate axes, start at the origin and draw the vector \( \mathbf{u} \). Since \( \mathbf{u} = \mathbf{j} + 2\mathbf{k} \), draw a vector 1 unit in the positive y-direction and 2 units in the positive z-direction.

Step 3: Draw Vector v

On the same coordinate axes, start at the origin and draw the vector \( \mathbf{v} \). Since \( \mathbf{v} = \mathbf{i} \), draw a vector 1 unit in the positive x-direction.

Step 4: Compute Cross Product u × v

Use the formula for cross product: \( \mathbf{u} \times \mathbf{v} = (a_2b_3 - a_3b_2)\mathbf{i} + (a_3b_1 - a_1b_3)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k} \) where \( \mathbf{u} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \) and \( \mathbf{v} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \). Here, \( \mathbf{u} = 0\mathbf{i} + 1\mathbf{j} + 2\mathbf{k} \) and \( \mathbf{v} = 1\mathbf{i} + 0\mathbf{j} + 0\mathbf{k} \). Simplifying, we get \( \mathbf{u} \times \mathbf{v} = 0\mathbf{i} + 2\mathbf{j} - 1\mathbf{k} \).

Step 5: Draw Cross Product u × v

Draw the vector \( \mathbf{u} \times \mathbf{v} = 2\mathbf{j} - \mathbf{k} \) starting at the origin. This means a vector 2 units in the positive y-direction and 1 unit in the negative z-direction.

Key Concepts

Cross Product

The cross product is a mathematical operation used to find a vector that is perpendicular to two given vectors in three-dimensional space. If you have two vectors \( \mathbf{u} \) and \( \mathbf{v} \), their cross product is denoted as \( \mathbf{u} \times \mathbf{v} \). This operation is particularly useful in physics and engineering because it helps to determine orientation and rotational effects.

• Formula: The formula for the cross product of two vectors \( \mathbf{u} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{v} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} \) is given by: \[ \mathbf{u} \times \mathbf{v} = (a_2b_3 - a_3b_2)\mathbf{i} + (a_3b_1 - a_1b_3)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k} \]
• Components: Notice how each component (\( \mathbf{i}, \mathbf{j}, \mathbf{k} \)) of the resulting vector derives from interactions of two components at a time,effectively using a determinant from linear algebra to calculate each substantial result.
• Result: The resulting vector from the cross product always points in a direction perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \).

Remember, the magnitude of the cross product is given by the area of the parallelogram that the two vectors span, calculated as \( |\mathbf{u} \times \mathbf{v}| = |\mathbf{u}| |\mathbf{v}| \sin \theta \), where \( \theta \) is the angle between \( \mathbf{u} \) and \( \mathbf{v} \).

3D Coordinate System

Understanding the 3D coordinate system is essential for working with vectors in spatial analysis. This system is composed of three axes: the x-axis, y-axis, and z-axis. These axes are perpendicular to each other and intersect at a point called the origin, often labeled as \((0, 0, 0)\).

• X-Axis: Extends horizontally, where positive values move to the right and negative to the left.
• Y-Axis: Extends vertically, where positive values move upwards and negative downwards.
• Z-Axis: Represents depth, coming out of or going into the page. Positive values move out towards the observer, and negative values move away.

A point or vector in this space can be described with coordinates \((x, y, z)\). For vectors like \( \mathbf{u} \) and \( \mathbf{v} \) in the original problem, their position and direction are described with reference to these three axes.Visualizing this system in your mind or drawing it helps in understanding operations like the cross product and ensuring the correct placement of vectors.

Vector Components

Vector components in three-dimensional space describe how a vector can be broken down along the x, y, and z axes. Each vector is a combination of these directions.

• X-Component: Describes the length of the vector in the direction of the x-axis, often denoted as a multiple of \( \mathbf{i} \).
• Y-Component: Shows the vector's length along the y-axis and is expressed in terms of the unit vector \( \mathbf{j} \).
• Z-Component: Represents how far the vector extends in the z-direction, multiplied by \( \mathbf{k} \).

The vector \( \mathbf{u} = \mathbf{j} + 2 \mathbf{k} \) from the problem highlights that it has no x-component, a y-component of 1, and a z-component of 2. Similarly, \( \mathbf{v} = \mathbf{i} \) only has an x-component and lacks components in the y and z directions.Visualizing these components helps in both computation and geometrical representation, ensuring you're able to manipulate vectors effectively in applied problems.