Question

Given three points whose position vectors are $x\hat{i}+y\hat{j}+z\hat{k},\hat{i}+z\hat{j}$ and $\hat{i}-\hat{j}$ find the condition for the points to be collinear,

Answer 1

Given,

$\begin{array}{c}x\hat{i}+y\hat{j}+z\hat{k}\\ \hat{i}+z\hat{j}\\ \hat{i}-\hat{j}arecollinear.\\ So,\\ \left(\begin{array}{ccc}x& y& z\\ a_1& z& 0\\ 1& -1& 0\end{array}\right)=0\\ x\left(0\right)-y\left(0\right)+z(-1-z)=0\\ \therefore z=0andz=-1\\ ifz=0andz=-1\\ so,forallx,ythesevectorsarecollinear.\end{array}$