two vectors of norm 1 that are orthogonal to the vectors v1=(-3,1,1,1), v2=(-1, 2, 2,0), and v3 = (1,0,3,-1)??

Question

can anyone help plzzz....

Answer 1

I didn't delve into the details. But now that you've got three vectors with 4 elements, it's usual that you've got 1 dimension left out, so that you should be able to derive one vector of 4 elements which are orthogonal to all 3 vectors. But two? seems not quite right, unless the three given vectors are co-linear, but they don't seem so.

Answer 2

Find two vectors of norm 1 that are orthogonal to the vectors v1 = <-3,1,1,1>, v2 = <-1, 2, 2,0>, and v3 = <1,0,3,-1>.
___________

First get the vectors, then normalize them. The dot product of the desired vectors with the given vectors will be zero. Let the desired vector be:

w =

w • v1 = • <-3,1,1,1> = -3a + b + c + d = 0

The other dot products are similar. So we have three equations in four unknowns.

-3a + b + c + d = 0
-a + 2b + 2c + 0d = 0
a + 0b + 3c + d = 0

Solve for a, b, and c in terms of d. And we get:

a = 2c
b = 0
d = 5c

Let c = 1 and the orthogonal vector to the three given vectors is:

w = = <2,0,1,5>
_________

Now normalize the vector.

|| w || = √(2² + 0² + 1² + 5²) = √(4 + 0 + 1 + 25) = √30

The two unit normal vectors are:

±w / || w || = <2,0,1,5>/√30 and <-2,0,-1,-5>/√30

Answer 3

For the moment, forget about the norm 1 part, and concentrate on the orthogonal part.

If x = (a,b,c,d) is such a vector, then the dot products with v1, v2, v3 are all zero:

-3a + b + c + d = 0
-a + 2b + 2c + 0d = 0
a + 0b + 3c -d = 0.

Note that for any scalar m, the vector mx will also be orthogonal. This implies that there should be infinitely many solutions, with a free variable. Thus, we can fix one of the variables. With that in mind, let us decree that d = 1. (With the last coordinate of v2 being 0, it is unlikely that the last coordinate of x will always be 0.)

Then
-3a + b + c = -1
-a + 2b + 2c = 0
a + 3c = 1.

Solve this system, for a, b, c. (If this has no solution, then d has to be zero, and do it again.)

With the vector (a,b,c,d) in hand, calculate
|x| = sqrt(a^2 + b^2 + c^2 + d^2) (<- the magnitude of the vector)

The two desired vectors will be x / |x| and -x / |x|.

Good luck.