Question Video: Finding the Ratio by Which a Point Divides a Line Segment Mathematics

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.