A positively charged particle projected towards the east is...

The magnetic force on a moving charge is given by $\vec{F} = q(\vec{v} \times \vec{B})$.

Given that the particle is positively charged ($q > 0$), the direction of the force is the same as the direction of $\vec{v} \times \vec{B}$.

Let the east direction be along the positive $x$-axis ($\hat{i}$), the north direction be along the positive $y$-axis ($\hat{j}$), and the upward direction be along the positive $z$-axis ($\hat{k}$).

The velocity of the particle is $\vec{v} = v\hat{i}$.

The magnetic force is $\vec{F} = F\hat{j}$.

Let the magnetic field be $\vec{B} = B_x\hat{i} + B_y\hat{j} + B_z\hat{k}$.

Substituting these into the force equation:

$F\hat{j} = q(v\hat{i} \times (B_x\hat{i} + B_y\hat{j} + B_z\hat{k}))$

$F\hat{j} = qv(B_y\hat{k} - B_z\hat{j})$

For the force to be directed towards the north ($\hat{j}$ direction), the $\hat{k}$ component must be zero and the $\hat{j}$ component must be positive.

This gives $B_y = 0$ and $-qvB_z > 0$.

Since $q$ and $v$ are positive, we must have $B_z < 0$.

A negative $z$-component indicates that the magnetic field must have a component directed downward. Thus, the magnetic field could be directed downward.