Question

The values of $\lambda$ such than $(x,y,z)\ne (0,0,0)$ and $(\widehat{i+}\hat{j}+3\hat{k})x+(3\hat{i}-3\hat{j}+\hat{k})y+(-4\hat{i}+5\hat{j})z=\lambda (x\hat{i}+y\hat{j}+z\hat{k})$ are

Answer 1

$\begin{array}{c}Giventhat\\ \left(\hat{i}+\hat{j}+\hat{k}\right)x+\left(3\hat{i}-3\hat{j}+\hat{k}\right)y+\left(-4\hat{i}+5\hat{j}\right)z=\lambda \left(x\hat{i}+y\hat{j}+z\hat{k}\right)\\ \Rightarrow \left(x+3y-4z\right)\hat{i}+\left(x-3y+5z\right)\hat{j}+\left(3x+y\right)\hat{k}=\left(\lambda x\hat{i}+\lambda y\hat{j}+\lambda z\hat{k}\right)\\ \Rightarrow x+3y-4z=\lambda x\Rightarrow \left(1-\lambda \right)x+3y-4z=0----\left(1\right)\\ x-3y+5z=\lambda y\Rightarrow x-\left(\lambda +3\right)y+5z=0-----\left(2\right)\\ and,\\ 3x+y=\lambda z\Rightarrow 3x+y-\lambda z=0-----\left(3\right)\\ now,\\ \left[\begin{array}{ccc}1-\lambda & 3& -4\\ 1& -\left(\lambda +3\right)& 5\\ 3& 1& -\lambda \end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\\ \Rightarrow \left[\begin{array}{ccc}1-\lambda & 3& -4\\ 1& -\left(\lambda +3\right)& 5\\ 3& 1& -\lambda \end{array}\right]=0\left[\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\ne \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\right]\\ 3x+4y=9---\left(1\right)\\ y=mx+1----\left(2\right)\\ Togetx-cordinateofpointofinter\sec tionof\left(1\right)\&\left(2\right)\\ 3x+4\left(mx+1\right)=9\\ \Rightarrow x=\frac{9-4}{3+4m}=\frac{5}{4m+3}\\ x-co-ordianteisinteger\\ 4m+3=-1,+1,-5,+5possible\\ take\\ 4m+3=-1\Rightarrow m=\left(-1\right)itistrue\\ 4m+3=1\Rightarrow m=\frac{-1}{2}itiswrong\\ 4m+3=-5\Rightarrow m=-2itistrue\\ 4m+3=+5\Rightarrow m=\frac{1}{2}itiswrong\\ so,intergervalueofmare\left(-1,-2\right)\\ 2integer\end{array}$