Vectors and Matrices problem

[malsch]

Hi,

I have the following situation: n = px + qy + rz = qi - 5qj - 11pk
where w, x, y and z are vectors and:
x = 4i + 2j - 3k
y = 5i - 3j + 8k
z = -2i - j + 4k

i need to find all the possible values of p, q and r.

I substituted x, y and z in the first equation and compared coefficients of i, j and k and ended up with the following equations:

4p + 4q - 2r = 0
2p + 2q - r = 0
8p + 8q +4r = 0

This implies that one solution is obviously 0 for all the 3 unknowns. I really do not know how to continue from here. i used gaussian elimination and ended up with the following matrix:

(4 4 -2 | 0)
(0 0 0 | 0)
(0 0 8 | 0)

Any help would be greatly appreciated. 10q

 

[MrAl]

Hi,

You already did some of the work. If we take your second equation and multiply it by 4 we get;
8p+8q-4r=0

and if we combine this with the last equation we have these two equations:
8p+8q-4r=0
8p+8q+4r=0

Adding these two we get:
16p+16q=0

so it becomes obvious that p and q must be negatives of each other, so pick a value for p and make q the negative of it. That should be any constant including zero.
That leaves r.
For the first equation we have:
4p+4q-2r=0

or with the above assumption we would have:
4k+4(-k)-2r=0

or simplified:
4k-4k-2r=0

which simplifies more to:
-2r=0

or simplified:
r=0

So it looks like if we call p K then we can call q -K and r=0.

 

[squishy36]

Your first two equations are linearly dependent. Thus, if there's a solution, it will depend on one of your variables.

 

[lebevti]

there are no solutions...the lines in 3D space are parallel to each other and dont intersect

 

[MrAl]

there are no solutions...the lines in 3D space are parallel to each other and dont intersect

Hi,

That's a very keen observation. Funny thing is though, the only way vectors can ever be equal is if they *ARE* parallel, so saying there are no solutions is kinda strange here. Lines are not vectors strictly speaking.

Also, do you guys read any replies before you yourself reply? I think it would pay to understand the replies already there before you bother to post. The only exception to this rule i think is when the thread is especially long and you dont want to have to take the time to read each and every post before you can add your own interesting reply. Other than that, maybe take a little time to read other posts first. I think it will help in the long run

 

[Ratchit]

malsch,

Were you looking for a consensus when you posted the identical problem here? **broken link removed**

Ratch

 

[Tesla23]

malsch,

Were you looking for a consensus when you posted the identical problem here? **broken link removed**

Ratch

and here:
http://forum.allaboutcircuits.com/showthread.php?t=63215