State the position vectors of the points with coordinates \((9,1,-1)\) and \((-4,0,4)\).

Imagine floating in space where you have complete freedom to move left or right, forward or backward, up or down. That's what it's like to navigate in 3D space. Coordinates in three-dimensional space are a natural extension of the familiar 2D Cartesian plane.

Just as on paper where you have an 'x' axis for horizontal movement and a 'y' for vertical, in 3D space, we add a 'z' axis for depth. This third axis allows us to precisely locate any point in space with a set of numbers called coordinates, typically written as \( (x, y, z) \).

When working with coordinates in 3D, we're dealing with a three-coordinate system where each point is unique to its location defined by its three respective numbers. For instance, the coordinates \( (9,1,-1) \) refer to a point that is 9 units along the 'x' axis, 1 unit along the 'y' axis, and -1 unit down the 'z' axis.

Vectors are the language of movement in space, explaining not just where you are but also how you can get from one point to another. They are mathematical objects that have both a magnitude (length) and a direction, typically denoted by an arrow over letters or boldface type. In written form, particularly in textbooks and homework assignments, we often use an ordered list of numbers called components to represent them.

Let's say we have a point in 3D space with coordinates \( (9,1,-1) \) and we want to find its position vector. We would start at the origin \( (0,0,0) \) and 'draw' an arrow to our point. The position vector is noted as \(\begin{pmatrix} 9 \ 1 \ -1 \/end{pmatrix}\) and uniquely represents the direction and magnitude of that arrow from the origin to the point in space.

In vector notation, we express the transition from one point to another using these column vectors, which serve as the coordinates of the terminal point when the initial point is the origin.

The Cartesian coordinate system is the grid-like structure that overlays 3D space, enabling us to map every location with a simple set of numbers. It is named after the French philosopher and mathematician René Descartes who invented this system.

Each axis of the Cartesian system is perpendicular to the other two, which simplifies the process of locating points in space. We usually call them the x-axis (horizontal), y-axis (vertical), and z-axis (depth). When we describe a point using Cartesian coordinates, we are specifying its distance from these three mutually perpendicular planes.

To better understand Cartesian coordinates, consider a room. Where the floor meets one wall is the origin. Moving along the floor away from that point defines the x-axis, moving up the wall is the y-axis, and moving deeper into the room is the z-axis. Where the three measurements meet in the air is the exact location of a point, given by its Cartesian coordinates. For our earlier example, the point \( (9,1,-1) \) would be 9 units from the wall along the floor (x), 1 unit up the wall (y), and 1 unit out from the floor into the room (z, but in this case, negative, meaning toward the origin if we consider 'into the room' as positive depth).