Let the vector be v = ai + bj + ck. Since the vector is perpendicular to the z-axis, its component along the z-axis must be zero. Thus, c = 0. The vector is v = ai + bj.
The vector makes equal angles with the x and y axes. This implies the components along the x and y axes must be equal in magnitude. Thus, a = b. The vector becomes v = ai + aj.
The magnitude of the vector is given as 3. Therefore: $$ \sqrt{a^2 + a^2} = 3 $$ $$ \sqrt{2a^2} = 3 $$ $$ a\sqrt{2} = 3 $$ $$ a = \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2} $$
Substituting the value of a back into the vector expression: $$ v = \frac{3\sqrt{2}}{2}\hat{i} + \frac{3\sqrt{2}}{2}\hat{j} $$
Final Answer: \frac{3\sqrt{2}}{2}\hat{i}+\frac{3\sqrt{2}}{2}\hat{j}
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