Three-Dimensional Vectors

Date: 01/29/2003 at 03:26:48
From: Reema
Subject: 3-dimensional vectors

I'm having a big problem with the idea of 3-dimensional vectors. I've 
been studying 2D vectors for a while, but am finding it very hard to 
understand and visualise the notion of a vector in 3 dimensions. I 
can't draw sufficient diagrams, nor envisage the situation in my head. 
Here's one example:

Four people are trying to pull a tree out of the ground. The first is 
pulling at an angle of 30 degrees to the horizontal with a force of 
200N. The 2nd person is 60 degrees clockwise around the stump from 
the first and is pulling at an angle of 56 degrees to the horizontal 
with a force of 155 newtons. The third is 75 degrees anticlockwise 
from the first and is pulling with 300N at 48 degrees to the 
horizontal. The fourth person is directly opposite the first and 
pulling with 300N, 45 degrees to the horizontal. Find the total force 
acting on the stump, and the lifting force.

Date: 01/29/2003 at 10:17:32
From: Doctor Rick
Subject: Re: 3-dimensional vectors

Hi, Reema.

This is a tough one to visualize! You need to break it down into 
several diagrams - projections of the vectors into different planes. 
For instance, imagine looking directly down on the stump from a great 
height. What you see won't involve the up/down dimension (which I'd 
call z) at all; you'll only see the x and y components of the vectors 
(which are parallel to the ropes). We have projected the vectors into 
the x-y plane.

I would first draw the projection in the x-y plane, and label each 
vector with the magnitude and the angle from the horizontal. I can't 
really draw the lengths of these vectors in this projection yet; the 
magnitudes of the vectors are not the lengths of their projections in 
the x-y plane. But at least I can get the angles (60 degrees 
clockwise, etc.), and I can write down all the information. You'll 
want to pick a direction for the first vector, say, along the positive 
x-axis. This establishes the coordinate system.

Now you'll need a separate projection for each force. You will be 
considering the vertical plane that contains the vector. For instance, 
for the second vector (155 N, 56 degrees to the horizontal), you'll 
draw a right triangle:

              *
             /|
            / |
      155 N/  |z
          /   |
         /    |
        /56   |
       *------+
           r

From this diagram you can find the z component of the force vector, 
and what I labeled as the r component of the vector. What is this 
component? It is the length of the vector in the x-y projection that 
I drew first. I can go back to that projection and label the length of 
the vector. From this (and the angle that I indicated in that diagram) 
I can determine the x and y components of the vector.

Once you have the x, y, and z components of each vector, you can add 
them component-wise. You're asked for the magnitude and the z 
component of the resultant force. 

If you had been asked for the direction (angle clockwise or 
anticlockwise from the first person, and angle from the horizontal), 
you'd want to work backward through the process by which we obtained 
the components. In the x-y projection, you can find the angle and the 
"r component" (that's cylindrical coordinate notation, by the way). 
Then transfer the r component to a new vertical-plane diagram for this 
vector, and use the r and z components to find the angle from the 
horizontal.

I hope this clarifies a hard-to-visualize problem. The usual trick in 
working with three dimensions, as I have shown, is to find a way to 
work in two dimensions at a time, choosing a plane that contains 
information you can use. Let me know if you'd like to discuss it 
further.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/