Is expiation obligatory upon one who swore to go with his brother to the beach to encourage him, but his brother went without informing him?

If the oath was uttered by your tongue unintentionally, it is considered a 'laghw' (idle) oath, and no expiation is required for it. However, if you intended it with your heart, it is a binding oath for which expiation becomes due if you break it. Since you broke your oath by not going with your brother, expiation is incumbent upon you, which is to feed ten poor people, or clothe them, or free a slave. If you are unable to do so, then you must fast for three days, and it is more cautious for them to be consecutive. His going without your knowledge has no effect, unless you intended to go with him if he informed you; in that case, expiation would not be binding upon you, because the intention of the one who swears the oath is what matters.