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.
Answer & explanation
Correct answer: option 4
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
$∴\vec v= 2\hat i-\hat j$