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

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


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?

Let’s call the position vector of the particle we’re looking for 𝑃. And if we write out 𝑃 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. We’re told we’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‘₯, 𝑦, and 𝑧 specify those three perpendicular directions. The coordinates of our point 𝑃 all match up, each one with a particular direction, the first with π‘₯, the second with 𝑦, and the third with 𝑧. If we plot our point 𝑃 on this set of axes, we know its π‘₯-value will be 1, its 𝑦-value will be negative 4.0, and its 𝑧-value will be positive 6.0. If we find the line along which π‘₯ equals one and 𝑦 is equal to negative four, then if we go out 6.0 units in the positive 𝑧-direction along that line, then we’ll find where the point 𝑃 is located.

In this exercise, we wanna solve for the position vector of that point 𝑃. We can draw the position vector 𝑃 in on our diagram as starting at the origin and going to the point 𝑃. For each of our three dimensions, π‘₯, 𝑦, and 𝑧, there is a corresponding unit vector, 𝑖 for π‘₯, 𝑗 for 𝑦, and π‘˜ for 𝑧. These unit vectors are the building blocks that we’ll use to create vectors in this three-dimensional space. With this correlation known, we can now write out the vector 𝑃 by referring to the coordinates of the point 𝑃.

The vector 𝑃 will have 1.0 unit in the 𝑖-direction. In the 𝑦-dimension, it will go negative 4.0 units, or negative 4.0𝑗. And along the 𝑧-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.

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