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

If 𝐢 ∈ 𝐴𝐡 and vector 𝐴𝐡 = 3 vector 𝐢𝐡, then 𝐢 divides vector 𝐡𝐴 by the ratio _. [A] 2 : 1 [B] 1 : 2 [C] 1 : 3 [D] 3 : 1

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Video Transcript

If 𝐢 is an element of 𝐴𝐡 and vector 𝐴𝐡 equals three times vector 𝐢𝐡, then 𝐢 divides vector 𝐡𝐴 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 𝐴𝐡, which we could model like this. We’re told that 𝐢 is an element of 𝐴𝐡, so that means that there will be a point 𝐢 somewhere on this line. If vector 𝐴𝐡 is equal to three times vector 𝐢𝐡, then that means that three of this length 𝐢𝐡 would make up the length of 𝐴𝐡. We could divide our length 𝐴𝐡 into three pieces, but the question is, is 𝐢 here or is 𝐢 here? If we consider if 𝐢 is at this lower point, then the length 𝐢𝐡 would look like this. But if we were to multiply 𝐢𝐡 by three, we wouldn’t get the length of 𝐴𝐡. We can then say that 𝐢 must be here, closer to 𝐡, as this length of 𝐢𝐡 would fit. Three lots of 𝐢𝐡 would give us 𝐴𝐡.

We now need to work out the question of how 𝐢 divides this vector 𝐡𝐴. We can then say if 𝐡𝐢 is one unit length long, then 𝐴𝐢 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 𝐡𝐴, so that means that we’re going from 𝐡 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 𝐴𝐡 instead, then that would’ve been the ratio two to one. But here, since 𝐢 is dividing vector 𝐡𝐴, then it’s the ratio one to two.

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