Answer
(a) Yes, it is possible. It occurs when the angle between ${\bf{u}}$ and ${\bf{v}}$ is $180^\circ $.
(b) ${\bf{u}}$ and ${\bf{v}}$ must be perpendicular vectors.
Work Step by Step
(a) By Eq. (4) of Theorem 3, the component of ${\bf{u}}$ along ${\bf{v}}$ is the vector ${{\bf{u}}_{||{\bf{v}}}}$:
${{\bf{u}}_{||{\bf{v}}}} = \left( {\frac{{{\bf{u}}\cdot{\bf{v}}}}{{{\bf{v}}\cdot{\bf{v}}}}} \right){\bf{v}}$.
If the angle between ${\bf{u}}$ and ${\bf{v}}$ is $180^\circ $ then ${\bf{u}}$ and ${\bf{v}}$ are in opposite direction.
We have
${{\bf{u}}_{||{\bf{v}}}} = \left( {\frac{{{\bf{u}}\cdot{\bf{v}}}}{{{\bf{v}}\cdot{\bf{v}}}}} \right){\bf{v}} = \left( {\frac{{ - ||{\bf{u}}||||{\bf{v}}||}}{{||{\bf{v}}|{|^2}}}} \right){\bf{v}} = - \frac{{||{\bf{u}}||}}{{||{\bf{v}}||}}{\bf{v}}$
Since ${\bf{u}}$ and ${\bf{v}}$ are in opposite direction, this implies that the component of ${\bf{u}}$ along ${\bf{v}}$ is in ${\bf{u}}$ direction.
Now, the component of ${\bf{v}}$ along ${\bf{u}}$ is the vector ${{\bf{v}}_{||{\bf{u}}}}$:
${{\bf{v}}_{||{\bf{u}}}} = \left( {\frac{{{\bf{v}}\cdot{\bf{u}}}}{{{\bf{u}}\cdot{\bf{u}}}}} \right){\bf{u}} = \left( {\frac{{ - ||{\bf{v}}||||{\bf{u}}||}}{{||{\bf{u}}|{|^2}}}} \right){\bf{u}} = - \frac{{||{\bf{v}}||}}{{||{\bf{u}}||}}{\bf{u}}$.
We conclude that the component of ${\bf{v}}$ along ${\bf{u}}$ have the opposite sign from the component of ${\bf{u}}$ along ${\bf{v}}$, or vice versa. So, the answer is yes, it is possible. It occurs when the angle between ${\bf{u}}$ and ${\bf{v}}$ is $180^\circ $.
(b) We have the component of ${\bf{u}}$ along ${\bf{v}}$ given by
${{\bf{u}}_{||{\bf{v}}}} = \left( {\frac{{{\bf{u}}\cdot{\bf{v}}}}{{{\bf{v}}\cdot{\bf{v}}}}} \right){\bf{v}}$
and the component of ${\bf{v}}$ along ${\bf{u}}$ given by
${{\bf{v}}_{||{\bf{u}}}} = \left( {\frac{{{\bf{v}}\cdot{\bf{u}}}}{{{\bf{u}}\cdot{\bf{u}}}}} \right){\bf{u}}$
If either of these two components is $0$, we must have ${\bf{u}}\cdot{\bf{v}} = {\bf{v}}\cdot{\bf{u}} = 0$ since ${\bf{u}}$ and ${\bf{v}}$ are nonzero vectors. Therefore, ${\bf{u}}$ and ${\bf{v}}$ must be perpendicular vectors.
