The $$x$$ and $$y$$ components of $$\vec A$$ are $$4\,m$$  and $$6\,m,$$  respectively. The $$x$$ and $$y$$ components of $$\left( {\vec A + \vec B} \right)$$  are $$10\,m$$  and $$9\,m$$  respectively. The magnitude of vector $${\vec B}$$ is:

Solution :
$$\overrightarrow A = 4\hat i + 6\hat j.$$   If $${B_x}$$ and $${B_y}$$ are the components of vector along $$x$$ and $$y$$ axes, then
$$\eqalign{ & 4 + {B_x} = 10, \cr & \therefore {B_x} = 6. \cr & {\text{Also}}\,6 + {B_y} = 9, \cr & \therefore {B_y} = 3. \cr & {\text{Thus,}}\,\,B = \sqrt {B_x^2 + B_y^2} = \sqrt {{6^2} + {3^2}} = \sqrt {45} \cr} $$