Question

If $\overrightarrow{a}$ is a unit vector such that $\overrightarrow{a}\times \hat{i}=\hat{j}$, find $\overrightarrow{a}.\hat{i}$

Answer 1

Let $\overrightarrow{a}=x\hat{i}+y\hat{j}+z\hat{k}$ where $\sqrt{x^2+y^2+z^2}=1$

then

$\overrightarrow{a}\times \hat{i}=\hat{j}$

$\begin{array}{c}\Rightarrow \left(x\hat{i}+y\hat{j}+z\hat{k}\right)\times \hat{i}=\hat{j}\\ \Rightarrow y\left(-\hat{k}\right)+z\left(\hat{j}\right)=\hat{j}\\ \Rightarrow -y\hat{k}+z\hat{j}=\hat{j}\\ \Rightarrow y=0,z=1\end{array}$

Now,

$\begin{array}{c}\sqrt{x^2+y^2+z^2}=1\\ \Rightarrow {\left[\sqrt{x^2+y^2+z^2}\right]}^2=1^2\\ \Rightarrow x^2+y^2+z^2=1\\ \Rightarrow x^2+0+1=1\\ \Rightarrow x=0\end{array}$

Therfore, $\overrightarrow{a}=\hat{k}$

So, $\overrightarrow{a}.\hat{i}=\hat{k}.\hat{i}=0$