91Ó°ÊÓ

In vector calculus, a vector representation is an essential concept that allows us to define and understand vectors in a three-dimensional space. A vector is typically represented as an arrow from one point to another, having both a direction and a magnitude.

In the given exercise, the vector \( \vec{A B} \) is the directed line segment from point \( A(4,0,-2) \) to point \( B(4,2,1) \). To represent this vector algebraically, we use the coordinates of points \( A \) and \( B \) to calculate the vector components. The algebraic representation of a vector provides a concise way of expressing what changes occur as we move from the starting point \( A \) to the endpoint \( B \).

Knowing how to represent vectors from different points is also essential. For instance, representing the vector from the origin (\( O(0,0,0) \)) allows easier analysis using standard formulas and operations in vector calculus.

Coordinate geometry, or analytic geometry, involves the study of geometric figures through a coordinate system. Understanding how vectors can be represented by coordinates in 3D space is crucial in this field.

In the exercise, points \( A \) and \( B \) are specified by their coordinates in a three-dimensional Cartesian plane: \( A(4,0,-2) \) and \( B(4,2,1) \). These coordinates allow us to plot the points and understand their spatial relationship.

Using coordinate geometry, we can calculate distances, angles, and other properties of geometric figures. Here, calculating the vector \( \vec{A B} \) from \( A \) to \( B \) showcases one of the practical applications of coordinate geometry. By plotting the vector's movement based on changes in \( x \), \( y \), and \( z \), we are employing the principles of coordinate geometry to understand spatial transformations.

Vector components break down a vector into its constituent parts along the coordinate axes, providing insight into its directional influence. Each component represents the vector's projection on an axis.

To find the components of the vector \( \vec{A B} \) in the exercise, we determine the change along each axis:

• In the x-direction: \( B_x - A_x = 4 - 4 = 0 \)
• In the y-direction: \( B_y - A_y = 2 - 0 = 2 \)
• In the z-direction: \( B_z - A_z = 1 - (-2) = 3 \)

Thus, the vector \( \vec{A B} \) can be expressed as \( (0, 2, 3) \), highlighting that its effect is solely in the y and z directions.

Understanding vector components is fundamental in physics and engineering, as it allows one to analyze real-world applications, like forces and motion along specific paths. Analyzing vector components simplifies complex problems into manageable, one-dimensional problems, enabling calculations like determining resultant forces or velocities.