Expert: Scotto - 3/29/2009

Question
1. Why does the vector equation of a plane have two parameters while the vector equation of a line only have one?

2. Why can two planes never intersect in a single point?

3. state whether the normals of the plane is collinear, coplanar, or neither.

3x - 4y +5z = 6
5x - 6y +7z = 8
6x - 8y +10z = 9
for this question, how do i know how to find out if it is coplanar or collinear? like what is the difference.

thanks for your help in advance!

Answer
Sam
1. Why does the vector equation of a plane have two parameters while the vector equation of a line only have one?


2. Why can two planes never intersect in a single point?

3. state whether the normals of the plane is collinear, coplanar, or neither.

3x - 4y +5z = 6
5x - 6y +7z = 8
6x - 8y +10z = 9
for this question, how do i know how to find out if it is coplanar or collinear? like what is the difference.

thanks for your help in advance!


1. One vector equation gives a line because one vector is moving in one direction.  Two vectors give a plane since you can move in to direction.   
One vector is like going straight in one direction.  If you only go in one direction, you’re walking on a line.
By two vectors they mean two directions.  Using two directions with different multiples, you can get to anywhere within a plane.

2. Think of two planes as two pieces of plywood that don’t bend at all.  They can touch at a single point on the edges because they are finite plains.  The planes they are talking about go on forever.  So if the two pieces of plywood are touching at the just a point on the edges, think about the planes touching like that.  Since planes go on forever, the contact between them would form a line.

3. If you look at the first and third equation, there is something to be noted.
Multiply the first equations 3 by 2 and you get the third equations 6.
Multiply the first equations –4 by 2 and you get the third equations –8.
Multiply the 5 by 2 and you get the 10.
So when the first equation is multiplied by 2, on the left side, you get equation 3.
Now multiply the right side.  That would be 6*2.  The answer is 12, and that’s not the 9 that is in the third equation.  Since the left side is multiplied by 2 in all the variables, these two equations can be said to define two parallel planes that never intersect.  Therefore there is no solution.  What can be said is the solutions to one and three are coplanar.

The equations have three variables, and that defines three dimensional space.
For each equation, the dimensions of that space are reduced by 1.
In other words, one equation makes the solution a 2 dimensional plane.
Two equations makes the solution a 1 dimensional line.
Three equations reduces it to one point.
That is, of course, if the solutions to the equations intersect.

As another example, consider if you had four equations.
The solution would be all of four dimensional space.
One equation would reduce it to everything in the space we understand (since 4-1=3).
Two equations would reduce it to a plane (since dimension is 4-2=2.
Three equations would reduce it to a line (since dimension is 4-3=1).

To take it even farther, suppose we had ten variables.  We won’t even try to define the space but to say it has ten dimensions.  If there were 8 intersecting constraints, we would have the solutions down to points in a plane, since 10-8 =2 and the dimensions of a plane are 2.

Basically, if you have n variables, you need n intersecting constraints to satisfy the equations.
If you have less than n constraints and most of them are inequalities, then you have on operations research problems frequently seen in insurance companies, supply of products to stores, supply lines to the front lines of a war, etc.

But I’ve said more than you probably want to know, so I hope the start did it for you.