Problem 8 - 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. \(A(4,0,-2), \quad B(4,2,1)\)

Step-by-Step Solution

Answer

The vector \( \vec{AB} \) is \((0, 2, 3)\).

Step 1: Identify the Coordinates of Points A and B

Point \( A \) is given as \( (4, 0, -2) \) and point \( B \) is given as \( (4, 2, 1) \). These points represent the initial and terminal points of the vector \( \overline{AB} \).

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

To find the vector \( \vec{AB} \), subtract the coordinates of point \( A \) from the coordinates of point \( B \). The vector is calculated as follows: \( \vec{AB} = (4 - 4, 2 - 0, 1 - (-2)) = (0, 2, 3) \).

Step 3: Draw Vector \( \vec{AB} \) from Point A

To visualize \( \vec{AB} \) starting from point \( A \), draw a vector that represents the movement from \( A(4,0,-2) \) to \( B(4,2,1) \) using the components \( (0,2,3) \).

Step 4: Represent Vector \( \vec{AB} \) Starting at the Origin

To draw an equivalent vector starting at the origin \((0,0,0)\), simply use the components of \( \vec{AB} \). From the origin, draw the vector \((0, 2, 3)\). This will provide you with the same directional and magnitude properties as the original vector \( \overline{AB} \).

Key Concepts

Coordinates of Points

In the context of vectors, understanding the coordinates of points is fundamental. Coordinates tell us the exact position of a point in the space. Each point is represented by a tuple, such as \( A(4, 0, -2) \) or \( B(4, 2, 1) \).
These tuples denote the position along each axis in a three-dimensional space. Here, the first number refers to the x-coordinate, the second to the y-coordinate, and the third to the z-coordinate.
Knowing the coordinates helps to define a vector's starting and ending points. This is crucial for calculating a vector's direction and magnitude, which are the core properties of vectors.

Vector Components

Vector components represent the change or movement along each axis when moving from one point to another. To find the components of vector \( \vec{AB} \), you subtract the coordinates of the initial point \( A \) from the coordinates of the terminal point \( B \).

For example, with \( A(4, 0, -2) \) and \( B(4, 2, 1) \), the vector components are calculated as follows:

• x-component: \( 4 - 4 = 0 \)
• y-component: \( 2 - 0 = 2 \)
• z-component: \( 1 - (-2) = 3 \)

Thus, the vector \( \vec{AB} \) is represented by \( (0, 2, 3) \). These components tell us that the vector has no change in the x-direction, moves up by 2 units in the y-direction, and up by 3 units in the z-direction.
Vector components make it easier to visualize the vector's direction and are crucial for graphing or manipulating vectors within coordinate systems.

Origin Representation

Origin representation of a vector simplifies analysis and allows for easy comparison among vectors. In this approach, vectors are redrawn to start at the origin point \( (0, 0, 0) \), while maintaining their original direction and magnitude.

From our given problem, the vector \( \vec{AB} \) from point \( A \) to point \( B \) is \( (0, 2, 3) \). By simply plotting this vector starting at the origin, you generate an equivalent vector. This equivalent vector shares the same direction and magnitude as the original, and its representation is straightforward:

• Starting point: Origin \( (0, 0, 0) \)
• Vector direction and components: \( (0, 2, 3) \)

Using origin representation is beneficial as it standardizes vectors, making them easier to compare, add, and manipulate in computations. It provides a consistent way of handling vector mathematics in three-dimensional space.