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

Step-by-Step Solution

Answer

Plot \( \mathbf{u} = \langle 1, 0, -1 \rangle \), \( \mathbf{v} = \langle 0, 1, 1 \rangle \), and \( \mathbf{u} \times \mathbf{v} = \langle 1, -1, 1 \rangle \).

Step 1: Understanding Vectors

First, we need to understand the given vectors \( \mathbf{u} \) and \( \mathbf{v} \). The vector \( \mathbf{u} = \mathbf{i} - \mathbf{k} \) can be written as \( \langle 1, 0, -1 \rangle \) indicating it points 1 unit in the positive x-direction and 1 unit in the negative z-direction. Similarly, the vector \( \mathbf{v} = \mathbf{j} + \mathbf{k} \) translates to \( \langle 0, 1, 1 \rangle \) indicating it points 1 unit in the positive y-direction and 1 unit in the positive z-direction.

Step 2: Sketching Coordinate Axes

Draw the 3D coordinate system with the x, y, and z-axes. The positive x-axis points forward, the positive y-axis points to the right, and the positive z-axis points upward.

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

Place the tail of \( \mathbf{u} \) at the origin \( (0, 0, 0) \). From the origin, mark a point at \( (1, 0, -1) \) in the coordinate space and draw an arrow from the origin to this point, representing the vector \( \mathbf{u} \).

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

Similarly, start at the origin \( (0, 0, 0) \). From here, go to the point \( (0, 1, 1) \) in the coordinate system to position the vector \( \mathbf{v} \). Draw an arrow from the origin to this point for the vector \( \mathbf{v} \).

Step 5: Calculating Cross Product \( \mathbf{u} \times \mathbf{v} \)

To find \( \mathbf{u} \times \mathbf{v} \), use the determinant method for cross products: \[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 0 & -1 \ 0 & 1 & 1 \end{vmatrix} = (0 - (-1))\mathbf{i} - (1 - 0)\mathbf{j} + (1 - 0)\mathbf{k} = \mathbf{i} - \mathbf{j} + \mathbf{k} \]. So, \( \mathbf{u} \times \mathbf{v} = \langle 1, -1, 1 \rangle \).

Step 6: Plotting Cross Product \( \mathbf{u} \times \mathbf{v} \)

Plot the vector \( \mathbf{u} \times \mathbf{v} \) starting from the origin and pointing to the coordinate \( (1, -1, 1) \). This shows the result of the cross product as the vector perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \).

Key Concepts

Cross Product

The cross product is a crucial operation in vector calculus that yields a vector perpendicular to two given vectors in a three-dimensional space. When calculating the cross product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \), symbolized as \( \mathbf{u} \times \mathbf{v} \), it results in a new vector that is orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \). This process is particularly important in physics and engineering for finding vectors that represent rotational forces and areas perpendicular to surfaces.

The calculation involves a determinant method. With vectors \( \mathbf{u} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{v} = \langle b_1, b_2, b_3 \rangle \), the cross product \( \mathbf{u} \times \mathbf{v} \) is given by the determinant:

• \( \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} \).
• This expands to \((a_2b_3 - a_3b_2)\mathbf{i} - (a_1b_3 - a_3b_1)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k}\).

Practically, for vectors \( \mathbf{u} = \langle 1, 0, -1 \rangle \) and \( \mathbf{v} = \langle 0, 1, 1 \rangle \), their cross product \( \mathbf{u} \times \mathbf{v} = \langle 1, -1, 1 \rangle \). Thus, this vector is perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \), positioned in 3D space as explained in the exercise solution.

Understanding the cross product is essential for applications that need to consider orientations and angles between vectors.

3D Coordinate System

In mathematics and physics, the 3D coordinate system is a framework used to define the position of points in three-dimensional space. The space is defined by three perpendicular axes, named the x-axis, y-axis, and z-axis, each representing one of the three dimensions.

Visualizing this coordinate system can help us comprehend spatial relationships between vectors and their components. In the 3D coordinate system:

• The x-axis usually points forward or towards the observer.
• The y-axis extends to the right.
• The z-axis proceeds upwards.

When sketching vectors within this space, one starts with the origin point, which is positioned at \( (0, 0, 0) \). Vectors such as \( \mathbf{u} \) and \( \mathbf{v} \) from the example can be plotted easily when understood in this framework, providing a visual representation of their components:

• Vector \( \mathbf{u} = \mathbf{i} - \mathbf{k} \), translates to \( \langle 1, 0, -1 \rangle \), indicating movement in the x-axis and negative z-axis.
• Vector \( \mathbf{v} = \mathbf{j} + \mathbf{k} \), translates to \( \langle 0, 1, 1 \rangle \), moving along the y-axis and in the positive z-axis.

Understanding this system is foundational for visualizing mathematical concepts such as vectors and for constructing more complex geometrical structures in 3D space.

Vector Representation

Vectors are mathematical entities with both magnitude and direction. They are often represented as arrows in coordinate systems and can be described using unit vectors. In three-dimensional space, vectors are commonly expressed in terms of the unit vectors \( \mathbf{i}, \mathbf{j}, \) and \( \mathbf{k} \), which represent directions along the x, y, and z axes respectively.

The representation as \( \mathbf{u} = \langle a, b, c \rangle \) provides a clear view of a vector's orientation and magnitude in 3D space.

• For \( \mathbf{u} \), each component \( a, b, \) and \( c \) specifies how far the vector extends in the direction of its respective axis.
• Vector \( \mathbf{u} = \mathbf{i} - \mathbf{k} \) is denoted as \( \langle 1, 0, -1 \rangle \), pointing along the x-axis and opposite to the z-direction.
• Similarly, vector \( \mathbf{v} = \mathbf{j} + \mathbf{k} \) translates to \( \langle 0, 1, 1 \rangle \), combining directions in both the y and z axes.

Understanding vector representation is essential for solving spatial problems and for manipulating these mathematical objects in areas like physics where they are used to describe forces, velocities, and other vector quantities. The tips to represent these vectors graphically start with plotting their tails at the origin and using the components to determine where their heads point to in the 3D coordinate system. This understanding aids in performing calculations such as the cross product, crucial for perpendicular vector derivations.