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The equation of a plane perpendicular to a vector and passing through a given point? 
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#1
Mar208, 01:37 AM

P: 6

Hey guys; im studying calculus and i came across a problem for which the book does not have an answer...
how do i find the equation of a plane perpendicular to vector (2i+3j+4k) and passing though the point (1,2,3) ^^the numbers are some i just made up and no this is not a homework q....in fact i would be glad if any of you can kindly explain to me how this problem can be solved and/or if they can give me the link to some webpage where they have stuff about this thanks :) 


#2
Mar208, 01:54 AM

P: 15

The scalar product or dot product, allows you to tell if two vectors are orthogonal or perpendicular among other things. So if you can visualize that the vector [tex] (2, 3, 4)^{T} [/tex] must be orthogonal to every point in the plane then the plane through the origin with normal vector [tex] (2, 3, 4)^{T} [/tex] would be given by the equation [tex] 2x + 3y +4z = 0 [/tex]. This is evident because [tex] (2, 3, 4) \bullet ( x, y, z )^{T} = 2x + 3y +4z = 0 [/tex] implies [tex] (2, 3, 4)^{T} [/tex] is orthogonal to every vector in the plane. Now I leave it to you to figure out how to shift the plane so that it goes through the specified point.



#3
Mar208, 01:56 AM

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P: 26,157

Hint: All lines in that plane will be perpendicular to that vector! So draw the line through the origin, O, parallel to (2i+3j+4k). You want to find the point, F, on that line which is the foot of the perpendicular from (1,2,3), which we'll call P. Call F (2a,3a,4a). Then what is the condition for OF to be perpendicular to FP? 


#4
Mar208, 05:24 AM

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The equation of a plane perpendicular to a vector and passing through a given point?
If you are expected to be able to do a problem like this, you should already know two things:
1) A line in 3 dimensions can be written in parametric equations, x= At+ x_{0}, y= Bt+ y_{0}, z= Ct+ z_{0}, where (x_{0}, y_{0}, z_{0}) is point on the line and [itex]A\vec{i}+ B\vec{j}+ C\vec{k}[/itex] is parallel to the line. 2) A plane can be written as a single equation, A(x x_{0})+ B(y y_{0})+ C_{0}(z z_{0})= 0 where (x_{0}, y_{0}, z_{0}) is a point on the plane and [itex]A\vec{i}+ B\vec{j}+ C\vec{k}[/itex] is perpendicular to the plane. (My mistake, you don't really need to know (1) to do this problem!) 


#5
Mar208, 10:51 AM

P: 6

thank you everyone for your help and suggestions.!
i solved it....in fact im ashamed to see how easy it was... but even then i might not have found the answer if not for ur help! thanks again Ps: i really should stop doing math till 1 am it took me tens of mins to even get my mind around problems at that time while it took less than a min in the morning. 


#6
Mar208, 11:31 AM

P: 6

hehe ran into another prob as i was going through
how do i go about a problem that gives me 3 points of a triangle A,B,C [ A (x1,y1), B(x2,y2),C(x3,y3) ] and asks me to find the interior angles of it? how should i start on this problem?(this is in the section that we use dot product) 


#7
Mar208, 11:39 AM

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P: 39,363

Have you learned that, in additon to the "add the products of the components" formula, [itex]\vec{u}\cdot\vec{v}= \vec{u}\vec{v} cos(\theta)[/itex], where [itex]\theta[/itex] is the angle between [itex]\vec{u}[/itex] and [itex]\vec{v}[/itex]? That should do it easily.



#8
Mar208, 05:09 PM

P: 6

hey , thanks :) i figured it out soon after i posted the q but was not able to delete my q right away cos i was away from the comp. but thanks again.



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