Q.Let $\bar b=4\hat i+3\hat j$ and $\bar c$ be two vectors perpendicular to each other in the XY-plane. Find the vector in the same plane having projection 1 and 2 along $\bar b$ and $\bar c$, respectively.

$\bar b=4\hat i+3\hat j$, $|\bar b|=5$, so $\hat b=\dfrac{4\hat i+3\hat j}5$. A vector perpendicular to

$\bar b$ in the XY-plane is $\bar c\propto-3\hat i+4\hat j$ (rotate $\bar b$ by $90^\circ$), so

$\hat c=\dfrac{-3\hat i+4\hat j}5$.

The required vector $\bar v$ has projection $1$ along $\bar b$ and $2$ along $\bar c$, so …

Find the unit vector along b, construct a perpendicular unit vector c in the same plane (by rotating b's components by 90 degrees), then com …

Using b and c themselves (not their unit vectors) as the basis for combining the given projection values, which …