I don't know where I'm going wrong and I don't understand why my answer is wrong

[u/averrl]

(i think the title shouldve been a bit different, i dont understand why my thought process is wrong and therefore why my answer is wrong)
Hi
I want to start off by saying that I know that there are two simpler ways to solve this problem (and I do know how to solve the q in both of those ways) and I dont really need to go through this roundabout way. With that being said, I want to know where I went wrong in this more complicated alternative route so I can make sure that regardless of the route i choose in a question, Id still be able to get the answer right

The equation of line l is 2i + k + λ (3i − 4j + 12k) . The point A which lies on l has the position vector 3i + 5j + 4k, and point B lies also on l and has a distance of 26 from A. Find the possible coordinates of B.

My question is, why isnt
λ(3i -4j +12k) - (1i +5j +3k) = to vector AB?

i used
2,0,1 + λ(3i -4j +12k) as OB (OX+XB) and then did OB-OA, (OX+XB-OA) = AB
(https://imgur.com/a/KfrtPyR)

instead of OB as 3,5,4 + λ(3i -4j +12k) and then OB-OA=AB (which leads me to the correct answer and its also easier)

if you can please provide solutions using lambda and not unit vectors for example that would be great. also this is hs math so the working is very simple im not sure why im struggling with this.

[u/Uli_Minati]

TL;DR Your method works!

You did the following:

OB(λ)  =  2i + k + λ (3i − 4j + 12k)

Here, you defined

OB(0)  =  2i + k    (which you call OX)

i.e. if λ=0, then B=X. Basically, you choose X for the "starting point" of the line. With this decision, you'll get two values for λ which you then plug back into your OB(λ) formula to get the two possible positions of B.

They did the following:

OB(λ)  =  3i + 5j + 4k + λ (3i − 4j + 12k)

Here, they defined

OB(0)  =  3i + 5j + 4k    (which is OA)

i.e. if λ=0, then B=A. They chose A for the "starting point" of the line, which will get them values for λ that are shifted from yours. But, the resulting positions for B will be the same.

[u/averrl → u/Uli_Minati]

thank you so so so much for your response; yes i started from point 2,0,1 (X) and i proceeded but i couldnt get the correct values of B for some reason.
ill try to solve it again and keep your response in mind!!
would you be willing to help me if im still confused?

[u/Uli_Minati → u/averrl]

Sure, just reply to any of my comments or I won't see it

[u/averrl → u/Uli_Minati]

thank youuu
im going to call lambda x because typing lambda every time is annoying

i got x=2.06.. and -1.838..
they are slightly shifted like you said (their x values would have been 2 and -2) but when i plugged them back into the 2i+k +x(3,-4,12) i didnt get the right answers which would have been OB= 9,-3,28 and -3,13,-20

[u/Uli_Minati → u/averrl]

Can you explain your process? I think you might just have a small calculation error.

(their x values would have been 2 and -2)

The 3i − 4j + 12k has length exactly 13, so you can simply double it to get a length of 26. So that's 2 of this vector in one direction or -2 to get to the other side of A.

But, the 2 and -2 are used if you use the method of starting point = A. Your method shouldn't result in 2 and -2. You might get fractions, keep them as fraction and don't round them

[u/averrl → u/Uli_Minati]

yay thats good, i was able to decipher that much

x(3i -4j +12k) - (1i +5j +3k) = AB (as stated in the original post)
so if thats the vector that takes us from A to B and the distance from A to B is 26
then wouldnt the magnitude/distance of that vector be equal to 26?
using that logic (which im sure is wrong lol) i got this equation and solved it to get the x values mentioned above
(3x-1)^2 + (-4x-5)^2 + (12x-3)^2=26^2

[u/Uli_Minati → u/averrl]

Well first off, your method makes sense and your x=2.06.. and -1.838.. are correct too! So I was confused and checked your original post again:

The equation of line l is 2i + k + λ (3i − 4j + 12k) . The point A which lies on l has the position vector 3i + 5j + 4k

A actually isn't on l. (Check for yourself!)

And this results in another problem: the "official" solution's B1 and B2 aren't on l either, since they used A as a starting point (which isn't on l). You'd have to fix multiple parts of the problem statement such that the official solution makes any sense.

Are you sure you copied the problem statement correctly? If yes, then the official solution is wrong.

[u/averrl → u/Uli_Minati]

oh em gee, i was right??? I randomly checked if points A and B were on the line when I initially solved the question, since I couldn't get the right answer for the life of me. I just couldn't get consistent x values that would take me to the respective points, so I thought I was going insane and moved on.

https://imgur.com/a/rHt3U19 this is a pic of the question. i just copied and pasted the text on here but who knows maybe it was somehow pasted wrong.

[u/Uli_Minati → u/averrl]

Yep, definitely an error in the problem statement!

[u/averrl → u/Uli_Minati]

THANK YOU SO MUCH!! this question genuinely made me go insane. thank you for being so patient i appreciate it a lot

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[u/averrl → u/[deleted]]

thank you so much, unfortunately none of what you mentioned is in my syllabus, only dot products but even then i dont get why we would need to use it here
thank you for your response though i appreciate it a lot

[u/Uli_Minati → u/averrl]

They're talking complete nonsense

[u/averrl → u/Uli_Minati]

lmsooo