Problem 7 - Essential Calculus Early Transcendentals

Question

Find a vector a with representation given by the directed line segment \(\overline{A B} .\) Draw \(\vec{A B}\) and the equivalent representation starting at the origin. \(A(0,3,1), \quad B(2,3,-1)\)

Step-by-Step Solution

Answer

The vector \(\vec{AB}\) is \((2, 0, -2)\). Start it at the origin as \((2, 0, -2)\).

Step 1: Understanding the Problem

We are asked to find the vector \(\vec{AB}\) given two points \(A(0,3,1)\) and \(B(2,3,-1)\). This vector is described by the directed line segment from point \(A\) to point \(B\).

Step 2: Calculate the Components of Vector \(\vec{AB}\)

The vector \(\vec{AB}\) can be calculated by taking the difference of the corresponding components of \(B\) and \(A\). Specifically: \[ \vec{AB} = (B_x - A_x, B_y - A_y, B_z - A_z) = (2 - 0, 3 - 3, -1 - 1) = (2, 0, -2) \]

Step 3: Representing \(\vec{AB}\) as a Vector Starting at the Origin

A vector can be represented equivalently starting at the origin by using the same components calculated in the previous step. So the vector \(\vec{AB}\) starting from the origin is \( (2, 0, -2) \).

Step 4: Drawing Vector \(\vec{AB}\)

To draw the vector \(\vec{AB}\), you can represent it as originating from point \(A\) and pointing towards \(B\), along the components calculated: \(2\) units along the x-axis, \(0\) units along the y-axis, and \(-2\) units along the z-axis from \(A\). As a vector from the origin, you can draw an arrow from \( (0, 0, 0) \) to \( (2, 0, -2) \).

Key Concepts

Vector Representation

Understanding vectors is crucial in vector calculus. A vector represents a quantity that has both magnitude and direction. Think of it like an arrow pointing from one place to another. In our exercise, the vector is described by the directed line segment \( \overline{A B} \) between points \(A(0,3,1)\) and \(B(2,3,-1)\).
This vector can be expressed by the components \((x,y,z)\) that indicate its length and direction along each axis in space. Thus, a vector can be seen as a way to efficiently capture and manipulate spatial information.

• **Origin Point**: The point where the vector begins (e.g., point \(A\)).
• **Terminal Point**: The point where it ends (e.g., point \(B\)).
• **Components**: Derived by the differences in positions of the terminal point and origin point.

This component-based representation is highly effective in mathematics and physics, as it allows us to break down complex motion or structural problems into manageable parts.

3D Vectors

Three-dimensional vectors occur in 3D space and are essential when dealing with real-world scenarios that require understanding of length, width, and height. Imagine the vector as an arrow that floats freely in space. Its direction is defined by shifts along the x, y, and z axes. In the exercise, point \(A\) and point \(B\) reside in 3D space and the vector \(\vec{AB}\) is defined by moving from \(A\) to \(B\).
The components found are \((2,0,-2)\):

• **X-component (2)**: Move 2 units along the x-axis.
• **Y-component (0)**: No movement along the y-axis.
• **Z-component (-2)**: Move -2 units along the z-axis, which is downward in standard coordinate systems.

Visualizing these vectors in 3D helps us understand complex physical phenomena and solve engineering problems, especially when visualizing forces, velocities, and accelerations in multiple dimensions.

Vector Components

To understand what components are, consider the vector \(\vec{AB}\) calculated as \((2,0,-2)\). Vector components are the projections of a vector along the x, y, and z axes. They essentially break down the vector into simple parts that determine how far the vector moves in each direction.
Calculating these components involves simple subtraction:

• Subtract the starting point's x value from the ending point's x value to get the x-component.
• Do the same for the y and z components.
• This method allows for systematic determination of displacements or any changes in position between two points.

These components allow vectors to be manipulated and understood in mathematical computations easily. This concept is critical when performing operations such as vector addition, scalar multiplication, and when dealing with forces in physics. Breaking down vectors into components makes it easier to apply mathematical formulas and visualize how different quantities interact spatially.