Hello there is a mathematical solution but we can solve this problem by just imagining a simple case.

assume a vector in the direction of negative x axis and B and C vector are making theta angle with the positive direction of x-axis and are symmetrical with the x axis therefore from the given condition we can find that cos theta equal to 1/4.

now the required magnitude of a cross sea is the same value as that of sin theta and which is equal to a square root of 15 / 16