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 𝑥, 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 𝑥, the second with
𝑦, and the third with 𝑧. If we plot our point 𝑃 on this set
of axes, we know its 𝑥-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 𝑥
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,
𝑥, 𝑦, and 𝑧, there is a corresponding unit vector, 𝑖 for 𝑥, 𝑗 for 𝑦, and 𝑘
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 𝑘-direction. This is the vector that describes
the position of the particle.
