vectors again and again

- anonymous

vectors again and again

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- anonymous

use vector to find the point that lies two thirds of the way from point P(4,3,0) to Q(1,-3,3)

- Psymon

Use points P and Q to make a component vector. Find the magnitude of this vector. Then multply the magnitude of the vector by the appropriate amount so that the magnitude is 2/3 of the original.

- anonymous

ok so it will be
<-3,-6,3> then magnitude would be 3sqrt(2)

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## More answers

- anonymous

now i am confused

- anonymous

3 sqrt(6) ****

- Psymon

\[\sqrt{9+36+9}= \sqrt{54}=3\sqrt{6} \]Lol, you corrected it.

- anonymous

haha i noticed that error after

- Psymon

So now all we need to do is multiply this magnitude by 2/3.

- anonymous

2 sqrt(6)

- Psymon

I guess I was looking at the question. That is the distance from the point P to the point 2/3 of the way there. I guess I thought the question was different at first.

- anonymous

oh so you know how to solve it?

- Psymon

Guess not x_x I didnt read the question well enough. I could probably figure it out, but the others out there would knwo what to do immediately where I have to think, haha.

- anonymous

haha ok

- Psymon

Yeah, I have an answer, but who knows, lol.

- anonymous

i know the answer but not the way

- Psymon

If you got the answer, then id like to see. I wanna see how miserably far off I am xDD

- anonymous

(2,-1,2) is the answer but i don't know how to get there

- Psymon

Yeah, thats what I got x_x My way worked but I doubt its the proper method xD

- anonymous

not to worry, help me how you got there

- Psymon

Yeah, sure. Well, I did what we did at first, found the component vector of <-3.-6,3> I realized thattheres no reason to find the magnitude, all we need to do is multiply these 3 points by 2/3. So this gives us <-2, -4, 2>. Now this helps us get 2/3 of the distance, but we're still not at the spot of our original vector. In order to get back, I added back onto to our vector what I took away in order to get the component vector. We did
<1-4, -3-3, 3-0> So instead I added back 4, added back 3, added back 0. That way Im now lined up with where I was lined up originally. So added those figures to the 2/3 vector I have, I do <-2 + 4, -4 + 3, 2 + 0>, which gets us <2, -1, 2>
Not the best method, but it worked x_x

- ganeshie8

drawing wud help visualize

- Psymon

|dw:1378185395072:dw|

- ganeshie8

thats the oly method thats simple enough i guess,
basically we just did this here :- OP + 2/3 PQ
O is the origin

- anonymous

|dw:1378185557539:dw|

- ganeshie8

draw 2/3 PQ also
put the initial point at origin

- anonymous

|dw:1378185679578:dw|

- ganeshie8

next, add OP and 2/3 PQ using parallelogram law

- ganeshie8

that gives geometrically the required point

- anonymous

|dw:1378185732167:dw|

- Psymon

|dw:1378185626689:dw|

- anonymous

guys tbh, i am so confused, how do you know that op + 2/3 pq = the point 2/3 from p towards q

- Psymon

|dw:1378185801120:dw|

- ganeshie8

if you're clear till here, that is, getting the vector 2/3 PQ, and OP. then the question is, why adding these wud give the point 2/3 from p towards q, right ?
|dw:1378185874435:dw|

- Psymon

Actually, the reason I know is because of the running around we were doing with the magnitude business. Once we saw that 2/3 of the magnitude was just 2sqrt(6), I actually tested multiplying each point by 2/3 instead. Then I checked the magnitude of what I multiplied by 2/3 and found out I got the same magnitude anyway. So knowing that all I needed to do to get 2/3 of the distance was multiplied the points by 2/3, I went and used the component vector of <-3,-6,3> to do that. Of course once I get 2/3 of that vector I still have to put it back where I found it by returning it to its initial position xD

- anonymous

yes @ganeshie8 and 2/3 of PQ is i think you both are on same route somewhere...i am the one who is not getting you both

- ganeshie8

i get you, you're not seeing why the final point of OP+2/3PQ falls on PQ vector.
let me ask you a q, how you wud add OP and 2/3 PQ ?

- ganeshie8

lets add them head to tail... if parallelogram method is bit unclear hear..

- ganeshie8

|dw:1378186282030:dw|

- ganeshie8

to add head to tail,
we put that 2/3PQ vector at the tail of OP

- anonymous

|dw:1378186311626:dw|

- ganeshie8

so.. ?

- anonymous

|dw:1378186363788:dw|

- ganeshie8

thats right, the resultant vector is not in the direction of PQ. but who cares about that ? we found the point thats 2/3 away from P in the direction PQ eh ?

- Psymon

Well, I think I can show you how I stumbled upon my reasoning. Remember, first we found our component vector of <-3,-6,3> Then we found the magnitude of that, which was 3sqrt(6). Once we multiplied this result by 3, we got a magnitude (length) that was 2/3 of that by multiplying by 2/3 to get 2sqrt(6). So we know the distance between the point P and the point we want will be 2sqrt(6) distance away.
Randomly, just to check, I took our component vector and multiplied each point by 2/3 first, just to see if Id get the same result. So multiplying the component vector by 2/3 gave me <-2, -4, 2> So just to check the magnitudeof that:
\[\sqrt{4 + 4 + 16}= \sqrt{24}= 2\sqrt{6}! \]
Because of this experimenting, I saw that all I had to do to get a point that was 2/3 of the distance along the line was multiply the points by 2/3. So that's exactly what I did. Multiplying the component vector by 2/3 gave me that vector <-2, -4, 2> Okay, awesome. Problem is now im centered at the origin and I need to go back to point P. Well, I moved from point P to the origin by subtracing 4, 3, and 0. So this time I added back 4, 3, and 0. Adding 4, 3, and 0 gave me the point <2, -1, 2>
I know this is long, but I thought maybe explaining my whole thought process would help >.<

- anonymous

oh ya i gatchyu

- Psymon

Okies, cool :P If you got either of us then good xD

- anonymous

alright so the answer in easy way would be op + 2/3 pq which would get me 2/3 of pq

- ganeshie8

... which would get me 2/3 of pq from p
good :)

- anonymous

alright ty...you both deserve medals but i can give only one...why don't you guys share it

- ganeshie8

actully you deserve one too.. :)
my wording above is also incorrect. ive corrected below :-
alright so the answer in easy way would be op + 2/3 pq which would get me \(\text{the point that lies}\) 2/3 of pq from p

- anonymous

haha gotcha ty

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