### Video Transcript

If πΆ is an element of π΄π΅ and
vector π΄π΅ equals three times vector πΆπ΅, then πΆ divides vector π΅π΄ by the ratio
blank. Option (A) two to one, option (B)
one to two, option (C) one to three, option (D) three to one.

Thereβs quite a lot of information
in this question. But letβs start with the fact that
we have this line segment π΄π΅, which we could model like this. Weβre told that πΆ is an element of
π΄π΅, so that means that there will be a point πΆ somewhere on this line. If vector π΄π΅ is equal to three
times vector πΆπ΅, then that means that three of this length πΆπ΅ would make up the
length of π΄π΅. We could divide our length π΄π΅
into three pieces, but the question is, is πΆ here or is πΆ here? If we consider if πΆ is at this
lower point, then the length πΆπ΅ would look like this. But if we were to multiply πΆπ΅ by
three, we wouldnβt get the length of π΄π΅. We can then say that πΆ must be
here, closer to π΅, as this length of πΆπ΅ would fit. Three lots of πΆπ΅ would give us
π΄π΅.

We now need to work out the
question of how πΆ divides this vector π΅π΄. We can then say if π΅πΆ is one unit
length long, then π΄πΆ would be equivalent to two of these lengths. So do we write this ratio as two to
one or one to two? Well, the direction here is very
important. Weβre given the vector π΅π΄, so
that means that weβre going from π΅ to π΄. We can therefore give our answer
that itβs the ratio one to two, which is given in option (B). Note that if we had been given the
vector π΄π΅ instead, then that wouldβve been the ratio two to one. But here, since πΆ is dividing
vector π΅π΄, then itβs the ratio one to two.