Let $β=4\hat i+3\hat j$ and $\vec γ$ be two vectors perpendicular to each other in the XY-plane. Find all the vector in the same plane having the projections 1 and 2 along $\vec β$ and $\vec γ$ respectively.

$2\hat i-\hat j$

Let $\vec v=x\hat i+y\hat i$

$\vec β.\vec v=0$

$4x+3y=0$

$y=\frac{-4}{3}x$

$∴\vec v=x\hat i-\frac{4}{3}x\hat j$

$\vec v=\frac{x}{3}(3\hat i-4\hat j)=(3\hat i-4\hat j)$

$β=4\hat i+3\hat j$

Let $\vec v=x\hat i+y\hat j$

$1=\frac{\vec v.\vec β}{|\vec β|}$ & $2=\frac{\vec v.\vec v}{|\vec v|}$

$1=\frac{4x+3y}{5}$ & $2=\frac{3x+4y}{5}$

∴$\begin{matrix}4x+3y=5\\3x-4y=10\end{matrix}$ solve

x = 2, y = -1

$∴\vec v= 2\hat i-\hat j$