# Orthogonal Vectors

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• December 2nd 2006, 07:55 AM
Bloden
Orthogonal Vectors
Let a_1 = [-6,-1,3,6,2,-3,-2,1],
a_2 = [-3,2,6,-3,-1,6,-1,2],
a_3 = [6,1,3,6,2,3,2,1],
a_4 [1,-6,-2,-1,3,2,-3,6]

Where each of the above are vectors (8x1)

1.) Show that the above vectors are orthogonal and then normalize them such that they all have length one. Call the new length-1 vectors u_1, u_2, u_3, u_4, respectively.

2.) Let U the 8x4 matrix which has the columns u_1, u_2, u_3, u_4. And, define:

y = [1,1,1,1,1,1,1,1], p = UU^(T)y, and z = y - p

Show that z is in-fact orthogonal to p.

My work:

#1 was easy; I showed the dot product between a_1 and a_2 is 0, between a_1 and a_3 is 0, etc etc.

I found that the length (norm) of all the vectors is 10; thus, in order to normalize them, I took the reciprocal and multiplied it by each of the vectors, to get u_1, u_2, u_3, u_4 as:

u_1 = [-3/5, -1/10, 3/10, 3/5, 1/5, -3/10, -1/5, 1/10]
u_2 = [-3/10, 1/5, 3/5, -3/10, -1/10, 3/5, -1/10, 1/5]
u_3 = [3/5, 1/10, 3/10, 3/5, 1/5, 3/10, 1/5, 1/10]
u_4 = [1/10, -3/5, -1/5, -1/10, 3/10, 1/5, -3/10, 3/5]

It's #2 that im stuck on.
• December 2nd 2006, 08:52 PM
Bloden
Can someone give me a start:

y = [1,1,1,1,1,1,1,1], p = UU^(T)y, and z = y - p

I dont really understand what the U and the transpose of U is for, then multiplying that by y...
• December 2nd 2006, 10:59 PM
earboth
Quote:

Originally Posted by Bloden
Can someone give me a start:

y = [1,1,1,1,1,1,1,1], p = UU^(T)y, and z = y - p

I dont really understand what the U and the transpose of U is for, then multiplying that by y...

Hello,

according to the text of your problem U is a 8x4-matrix. I'll use decimal numbers:

$U=\left(\begin {array}{cccc}-0.6&-0.3&0.6&0.1 \\-0.1&0.2&0.1&-0.6 \\0.3&0.6&0.3&-0.2 \\ 0.6&-0.3&0.6&-0.1 \\0.2&-0.1&0.2&0.3 \\-0.3&0.6&0.3&0.2 \\-0.2&-0.1&0.2&-0.3 \\0.1&0.2&0.1&0.6 \end {array} \right)$

Now calculate the transposed(?) matrix multiply both by the vector y.

EB
• December 3rd 2006, 02:52 PM
Bloden
Thanks Eartboth.

I used Maple and created U and U^(T);

I multiplied them together and got a 4x4 matrix:

[[41/50, 0, 0, 0], [-9/100, 1, 0, 0], [9/50, 0, 1, 0], [3/100, 0, 0, 1]];

Now, I have to multiply this by vector y; but how do I multiply a 4x4 matrix by an 8x1 vector...

Once I get this value, I wil lthen have to take y and subtract this...how is this going to show z orthogonal to p? I'm assuming I will use dot product to multiply z to p and see that they're equal to 0.