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# c4 vectors extra watch

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1. C4 - 5K question 12
I've done a,b and c, but i'll put down the whole question as its linked (I think).
Refered to a fixed origin O, the points A,B,C have position vecotrs (9i -2j +k), (6i +2j +6k) and (3i +pj +qk) respectively. (p/q constants)
a. Find the equation of the line l which goes through A and B.

b. given c is on the line, calculate p and q

c. calculate the acute angle between OC and AB

The point D lies on AB and is such that OD is perpendicular to AB.
d. Find the position vector of D.

Ok i've done a/b/c as said (after a bit of frustration and my only stupid mistakes) but now D has me stumped. Any help will be much appreciated and i'll try and will help me see where im going wrong in my working out. Cheers
2. (Original post by Daven)

The point D lies on AB and is such that OD is perpendicular to AB.
d. Find the position vector of D.
From a Previous part you found the equation of l which passes through A and B. r=(9-3t)i +(4t-2)j+(1+5t)K

We know D lies on that line so we can say D has position vector:
OD=(9-3t)i+(4t-2)j+(1+5t)K

To find the value of t:
For OD and AB to perpendicular;
OD.AB=0

(9-3t) (-3)
(4t-2) . (4) = 0
(1+5t) (5)

-3(9-3t)+4(-2+4t)+5(1+5t)=0
-27+9t-8+16t+5+25t=0
50t-30=0
t=3/5

Now sub this value for T into your point D to get exact position vector D.
OD=(9-3(3/5))i +(4(3/5)-2)j+(1+5(3/5))k
OD=36/5i + 2/5j +4k
3. Thanks, seems so simple :| I'll blame it on the long day

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