# Question Video: Determining the Position Vector given the Coordinates of a Point

The coordinates of a particle in a rectangular coordinate system are (1.0, β4.0, 6.0). What is the position vector of the particle?

02:30

### Video Transcript

The coordinates of a particle in a rectangular coordinate system are 1.0, negative 4.0, and 6.0. What is the position vector of the particle?

Letβs call the position vector of the particle weβre looking for π. And if we write out π as a point in space, that point is given as 1.0, negative 4.0, 6.0, where each number refers to a value along a dimension. Weβre told weβre working in a rectangular coordinate system. This is another name for the Cartesian system.

We can draw a three-dimensional coordinate system where π₯, π¦, and π§ specify those three perpendicular directions. The coordinates of our point π all match up, each one with a particular direction, the first with π₯, the second with π¦, and the third with π§. If we plot our point π on this set of axes, we know its π₯-value will be 1, its π¦-value will be negative 4.0, and its π§-value will be positive 6.0. If we find the line along which π₯ equals one and π¦ is equal to negative four, then if we go out 6.0 units in the positive π§-direction along that line, then weβll find where the point π is located.

In this exercise, we wanna solve for the position vector of that point π. We can draw the position vector π in on our diagram as starting at the origin and going to the point π. For each of our three dimensions, π₯, π¦, and π§, there is a corresponding unit vector, π for π₯, π for π¦, and π for π§. These unit vectors are the building blocks that weβll use to create vectors in this three-dimensional space. With this correlation known, we can now write out the vector π by referring to the coordinates of the point π.

The vector π will have 1.0 unit in the π-direction. In the π¦-dimension, it will go negative 4.0 units, or negative 4.0π. And along the π§-axis, it will move positive 6.0 units, or 6.0 in the π-direction. This is the vector that describes the position of the particle.