We consider the directed intermittent search for one or more targets in a one-dimensional domain with stochastic resetting. A particle (searcher) randomly switches between a stationary search phase and a rightward moving ballistic phase. The particle can detect a target at some fixed rate whenever it is within range of the target and is in the stationary state. In the absence of resetting, there is a nonzero probability of failure. We calculate the hitting (detection) probability and conditional mean first passage time (MFPT) with and without resetting, for both a single target and a pair of competing targets. We also present an alternative probabilistic method for taking into account the effects of resetting, which is based on conditional expectations, stopping times and an application of the strong Markov property. Such an approach has previously been used to analyze diffusion processes in randomly switching environments.