Find a vector that is perpendicular to two other 4 dimensional vectors

Question

I have the following problem:

"Given the two vectors $w_1=(5,0,1,1)$ and $w_2=(1,1,0,5)$.

Find a vector that is orthogonal to both $w_1$ and $w_2$."

I tried to create an augmented matrix, and reduce it to echelon form and this is what I got:

$$ F= \left(\begin{matrix}1&0&\frac{1}{5}&\frac{1}{5}&0\\0&1&\frac{-1}{5}&\frac{24}{5}&0\end{matrix}\right) $$

But how do I continue from this. I know I'm supposed to make a parametric form, but I don't know how to handle this matrix.

score 2 · Answer 1 · answered Dec 01 '19 at 15:26

2

All you have to do is to find a solution of the system$$\left\{\begin{array}{l}5a+c+d=0\\a+b+5d=0.\end{array}\right.$$One such solution is $(0,0,0,0)$, of course. If you're after a non-zero solution, you can take $(1,-1,-5,0)$, for instance

answered Dec 01 '19 at 15:26

José Carlos Santos

427,504

Ah, okay, it's really that simple... Thank you. – Carl Dec 01 '19 at 15:27
But if I were to make a parametric equation from that matrix, how would I then do it? – Carl Dec 01 '19 at 15:49
Solving my system, you get that $a=-\frac{c+d}5$ and that $b=\frac{c-24d}5$. Is this what you want? – José Carlos Santos Dec 01 '19 at 15:54
No. I mean the reduced row matrix I posted in the question. – Carl Dec 01 '19 at 15:57
1

@Carl See https://math.stackexchange.com/a/1521354/265466 for how to read a kernel basis from the rref. – amd Dec 01 '19 at 20:57

score 1 · Accepted Answer · answered Dec 01 '19 at 15:57

1

$(0,1,1)\times(1,0,5)=(5,1,-1)$ is perpendicular to $(0,1,1)$ and $(1,0,5)$,

so $(0,5,1,-1)$ is perpendicular to $(t,0,1,1)$ and $(u,1,0,5)$.

answered Dec 01 '19 at 15:57

J. W. Tanner

60,406

1

This is actually a brilliant answer. I have used this method quite a lot, but forgot to mark it as the answer. Thank you! – Carl Jan 17 '21 at 22:00

Find a vector that is perpendicular to two other 4 dimensional vectors

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