If u and v are perpendicular, then their dot product is zero:
[tex]\vec u \cdot \vec v = (5\,\vec\imath-4\,\vec\jmath+p\,\vec k) \cdot (2\,\vec\imath+t\,\vec\jmath - 3\,\vec k) = 10 - 4t - 3p = 0[/tex]
Solve for p :
[tex]10 - 4t - 3p = 0[/tex]
[tex]3p = 10 - 4t[/tex]
[tex]\boxed{p = \dfrac{10-4t}3}[/tex]
