April 7th, 2014, 10:27 AM  #1 
Member Joined: Apr 2014 From: liverpool Posts: 37 Thanks: 4  intersection of two lines (vectors)
I have been stuck on this question for a while now, can anybody help ? Vector equation of line L1=(9,2,1) + M(3,4,0) Vector equation of line L2=(2,3,2) + T(a,3,b) These lines are perpendicular and intersect at point P. Find unknowns a and b. Find point of intersection P. I have not came across one of these questions with two uknowns before, only one. Can anybody give me a hint into how to solve this ? Any help much appreciated. 
April 7th, 2014, 10:35 AM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs 
What is the criterion for orthogonal (or perpendicular) vectors?

April 7th, 2014, 10:41 AM  #3 
Member Joined: Apr 2014 From: liverpool Posts: 37 Thanks: 4 
That their scalar product is zero ? Or did you mean something else ?

April 7th, 2014, 10:52 AM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs 
The dot product of the two direction vectors must be zero. This will allow you to find $a$...what do you get?

April 7th, 2014, 10:54 AM  #5 
Member Joined: Apr 2014 From: liverpool Posts: 37 Thanks: 4 
i get a=4

April 7th, 2014, 11:00 AM  #6 
Member Joined: Apr 2014 From: liverpool Posts: 37 Thanks: 4 
i end up with a=4 t=1 b=3 m=1. point of intersection as 6,6,1 ? 
April 7th, 2014, 11:05 AM  #7 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Correct! So now we have: $\displaystyle \textbf{L}_1=\langle 9,2,1 \rangle+M\langle 3,4,0 \rangle$ $\displaystyle \textbf{L}_2=\langle 2,3,2 \rangle+T\langle 4,3,b \rangle$ Next, let's let point $P$ be $(x,y,z)$. Since the two lines intersect at that point, we may state: $\displaystyle x=3M+9=4T+2$ $\displaystyle y=4M+2=3T+3$ $\displaystyle z=1=bT+2$ You now have 3 unknowns in 3 linear equations...what do you find when you solve this system? 
April 7th, 2014, 11:08 AM  #8 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  
April 7th, 2014, 11:09 AM  #9 
Member Joined: Apr 2014 From: liverpool Posts: 37 Thanks: 4 
you must have missed the post i just sent, (6,6,1) ?

April 7th, 2014, 11:10 AM  #10 
Member Joined: Apr 2014 From: liverpool Posts: 37 Thanks: 4 
Thanks alot for your help ! really good advice !


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intersection, lines, vectors 
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