given vector: $u = <1, -7, 2>$ Find a vector $v$ and $w$ which are are orthogonal to each other and to $u$
I tried the following: $$ v \cdot u = <1,1,c> \cdot <1,-7,2> \rightarrow c= 3$$ $$ v = <1,1,3>$$ $$ Proof: v \cdot u = 0 \rightarrow True$$ For $w$, I used cross multiplication: $$\begin{array} c<1,-7,2> \\ <1,1,3> \end{array} $$ $$ w=<-23,-1,8> $$ When I try to prove it it keeps failing: $$Proof: v \cdot w= 0 \rightarrow True $$ $$Proof: u \cdot w= 0 \rightarrow False \rightarrow -14 $$
I tried looking for my mistake with no luck. I am sure there is a step I am not understanding correctly or I cannot see it after staring at it for such a long time. Can anyone look at this and let me know where I went wrong ?
Your mistake is in the final calculation of $u \cdot w$. We have $$u \cdot w = \langle 1, -7, 2 \rangle \cdot \langle -23, -1, 8 \rangle = 1(-23)+(-7)(-1)+2(8) =-23+7+16=0.$$