Video Transcript
If 𝐶 is an element of 𝐴𝐵 and
vector 𝐴𝐵 equals three times vector 𝐶𝐵, then 𝐶 divides vector 𝐵𝐴 by the ratio
blank. Option (A) two to one, option (B)
one to two, option (C) one to three, option (D) three to one.
There’s quite a lot of information
in this question. But let’s start with the fact that
we have this line segment 𝐴𝐵, which we could model like this. We’re told that 𝐶 is an element of
𝐴𝐵, so that means that there will be a point 𝐶 somewhere on this line. If vector 𝐴𝐵 is equal to three
times vector 𝐶𝐵, then that means that three of this length 𝐶𝐵 would make up the
length of 𝐴𝐵. We could divide our length 𝐴𝐵
into three pieces, but the question is, is 𝐶 here or is 𝐶 here? If we consider if 𝐶 is at this
lower point, then the length 𝐶𝐵 would look like this. But if we were to multiply 𝐶𝐵 by
three, we wouldn’t get the length of 𝐴𝐵. We can then say that 𝐶 must be
here, closer to 𝐵, as this length of 𝐶𝐵 would fit. Three lots of 𝐶𝐵 would give us
𝐴𝐵.
We now need to work out the
question of how 𝐶 divides this vector 𝐵𝐴. We can then say if 𝐵𝐶 is one unit
length long, then 𝐴𝐶 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. We’re given the vector 𝐵𝐴, so
that means that we’re going from 𝐵 to 𝐴. We can therefore give our answer
that it’s the ratio one to two, which is given in option (B). Note that if we had been given the
vector 𝐴𝐵 instead, then that would’ve been the ratio two to one. But here, since 𝐶 is dividing
vector 𝐵𝐴, then it’s the ratio one to two.
