Answer to Question #209798 in Linear Algebra for Faith

Question

1.let x<0.find the vector n=<x,y,z> that is orthogonal to all three vectors u=<1,1,-2>,v=<-1,2,0> and w=<-1,0,1>.

2.find a unit vector that is orthogonal to both u=<0,-1,-1> and v=<1,0,-1>.

Answer 1

(1) Let t be the fourth coordinate

For n to be orthogonal to the three vectors. Then,

$n=\begin{vmatrix} i & j &k &t\\ 1 & 1 &-2 &0\\ -1&2 &0 &0\\ -1& 0 &1 &0 \end{vmatrix}\\ n=t$

(2) The unit vector can be find by first getting the vector

$n_1=\begin{vmatrix} i & j &k \\ 0 & -1& -1\\ 1& 0& -1 \end{vmatrix}\\ n_1=i-j+k\\ \|n_1\|=\sqrt{1^2+(-1)^2+1^2}=\sqrt3\\ \widehat{n_1}=\frac{n_1}{\|n_1\|}\\ \widehat{n_1}=\frac{1}{\sqrt3}i-\frac{1}{\sqrt3}j+\frac{1}{\sqrt3}k\\$