 Linear algebra study realm  


Auteur  Message 

Sweet Admin
Messages : 270 Date d'inscription : 14/09/2007 Age : 25
 Sujet: Linear algebra study realm Dim 8 Mai 2011  15:13  
 Final exam study realm is henceforth open!!! Subscribe! Subscribe! 

 
spirit
Messages : 222 Date d'inscription : 25/09/2007 Age : 24
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  15:14  
 

 
JellyFish
Messages : 166 Date d'inscription : 18/09/2007 Age : 23
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  15:17  
 God, thank you! I need help now, immediately! How do you know what method to use when they ask questions like find a closest point from a plane, or find a line through a point and a plan and all those confusing questions? _________________ 

 
Sweet Admin
Messages : 270 Date d'inscription : 14/09/2007 Age : 25
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  15:29  
 Like... what kind of questions exactly. Because there are a lot of variants of them. 

 
spirit
Messages : 222 Date d'inscription : 25/09/2007 Age : 24
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  15:33  
 You see you can't even help her!
Let me answer this question, for I possess the ultimate power of algebra in my brain!
So, for your question: "How do you know what method to use when they ask questions like find a closest point from a plane, or find a line through a point and a plan and all those confusing questions?" You use the cross product when it comes to find an orthogonal vector and the dot product to deduct the closest point. 

 
JellyFish
Messages : 166 Date d'inscription : 18/09/2007 Age : 23
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  15:34  
 That's... So how exactly does it apply to all questions? _________________ 

 
Sweet Admin
Messages : 270 Date d'inscription : 14/09/2007 Age : 25
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  15:35  
 See, Spirit, you're not helping. 

 
spirit
Messages : 222 Date d'inscription : 25/09/2007 Age : 24
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  15:41  
 Oh yeah? Well, can you answer better? 

 
JellyFish
Messages : 166 Date d'inscription : 18/09/2007 Age : 23
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  16:00  
 Ok, let me make a list of questions, all in R3
1) Find the closest point to a line. 2) Find the closest point to a plane. 3) Find the closest distance between two planes. 4) Find the closest distance between two lines. 5) Find the closest distance between a point and a line. 6) Find the intersection of two planes. 7) Find the intersection of two lines. 8) Find the intersection of a line and a plane. 9) Find two orthogonal planes which intersect at a given line. 10) Find a plane which contains a given line. 11) Find a plane which contains a given line and which is parallel to a specific axis. 12) Find two parallel plane each containing one of the given skewlines. 13) Find a plane with contain a given point and a line. 14) Find a plane containing two intersecting lines. 15) Find a plane containing two parallel lines. 16) Find the line of intersection of two planes. 17) Find a plane that passes through the origin and orthogonal to two given planes. 18) Find a plane passing through a given point and parallel to a specific axis. 19) Find a plane which contains a given line and orthogonal to a given plane 20) Find two planes which intersect at a given point. 21) Find a line that is perpendicular to a given line.
_________________ 

 
spirit
Messages : 222 Date d'inscription : 25/09/2007 Age : 24
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  20:00  
 o_O There are so many  but no fear, I can handle all this. HOHO!  Spoiler:

Find the closest point to a line. 2 methods: Use the projection or the dot product. Suppose you have line: X = tu + A, a point P, and closest point on the line to P is Q The projection method would give you: Proj_u AP + A = Q. The dot product method gives: PQ*u=0, since Q is on the line, you can write Q in form of the line's equation.
Find the closest point to a plane. Suppose you have a plane: ax+by+cz = d and a point P. The closest point form the point to the plane is Q. Therefore, Q is the intersection of the line PQ and the plane. Line PQ can be written as (x,y,z) = t(a,b,c) + P. Substitute PQ equation into the plane's equation and solve for t. Then, you get the point.
Find the closest distance between two parallel planes. Provides that the two plane are parallel, thus they have the same normal. For two planes ax+by+cz = d1 and ax+by+cz = d2, the formula is d2d1/(a^2 + b^2 + c^2)^(1/2) = dist.
Find the closest distance between two lines. Suppose that you have a line X1 = tu + A and X2 = sv + B. Define two plane. Use u x v for their normal and find d1 and d2 with A and B. Then use d2d1/(a^2 + b^2 + c^2)^(1/2) = dist.
Find the closest point between two skewlines. Suppose you have a line X1 = tu + A = t(u1,u2,u3)T + (a1,a2,a3) which contains the point P that is the closest to Q on the line X2 = sv + B = s(v1,v2,v3)T + (b1,b2,b3)T. We know that PQ*v = PQ*u = 0, solve for s or t. P(tu1 + a1, tu2 + a2, tu3 + a3) Q(sv1 + b1, sv2 + b2, sv3 + b3) PQ = (sv1 + b1  tu1  a1, sv2 + b2  tu2  a2, sv3 + b3  tu3  a3)T (Let it be = (i,j,k)T PQ*v = iv1 + jv2 + kv3 = 0 PQ*u = iu1 + iu2 + iu3 = 0 2 equations, 2 unknowns, solve for s if you want to find Q, solve for t if you want to find P.
Find the closest distance between a point and a line. Suppose you have a line X= tu + A and a point P, use the formula: AP x u/u
Find the intersection of two planes. You have 2 equations, solve the system.
Find the intersection of two lines. Suppose, in the first line, you have x = at + d, y = bt + e, z = ct + f, and in the second line, you have x = 2as + d, y = 2bs + e, z = 2cs + f. Solve t or s. Let's say you want to solve for s, you get s = (xd)/2a, s = (ye)/2b and s = (zf)/2c. Substitute the first line into it, you get s = ((at + d)d)/2a, s = ((bt + e)e)/2b and s = ((ct + f)f)/2c See if the equation is consistent according to t. If if it is for t=k, such as k is a specific value, then the lines intersect at t=k. If if is consistent for all t, then they are the same line. If it's never consistent for any t, then they do not intersect.
Find the intersection of a line and a plane. Substitute the equation of the equation of the line into the plane. Similar to "Find the closest point to a plane."
Find two orthogonal planes which intersect at a given line. Use dot product. Suppose that the line is X = t(u1,u2,u3)T + (p1,p2,p3)T. The normal n = (a,b,c)T of the plane is orthogonal to the vector u = (u1,u2,u3)T. Thus, n*u = 0 = au1 + bu2 + cu3 = 0, Thus, n = s(u2/u1, 1, 0)T + t(u3/u1, 0, 1)T Take any n and use n x u to find another normal for the second plane. Use the point (p1,p2,p3)T to find d.
Find a plane which contains a given line. Just do the first part of "Find two orthogonal planes which intersect at a given line."
Find a plane which contains a given line and which is parallel to a specific axis. The normal of the plane is perpendicular from the axis and the vector of the line. Use the cross product and use the point on the line to find d. i.e.: Suppose that the line is X1 = tu + A and the axis is X2= sv The normal n = u x v Then substitute A to find d.
Find two parallel plane each containing one of the given skewlines. The normal of two planes is the same, use cross product of the two vectors in the lines. Use the points on each line to find each d. i.e.: Suppose that the lines are X1 = tu + A and X2 = sv + B. The normal n = u x v for both plane. Find d1 by using A and d2 by using B.
Find a plane with contain a given point and a line. Suppose that the line is X = tu + A and the point is P. The normal n = u x AP. Use A or P to find d.
Find a plane containing two intersecting lines. Suppose that the lines are X1 = tu + A and X2 = sv + B. Normal n = u x v and use A or B to find d.
Find a plane containing two parallel lines. Suppose that the lines are X1 = tu + A and X2 = sv + B. Since they are parallel, u = kv, thus we cannot use the cross product. Use normal n = u x AB or n = v x AB and use A or B to find d.
Find the line of intersection of two planes. You've already asked that. =.=
Find a plane that passes through the origin and orthogonal to two given planes. The normal of the questioning plane is orthogonal to the normals of the given planes, i.e. use cross product. Use the origin (0,0,0)T to find d.
Find a plane passing through a given point and parallel to a specific axis. Find the closest point A on the axis from the given point P. AP is the normal, use P to find d.
Find a plane which contains a given line and orthogonal to a given plane. Suppose that the line is X = tu + A and the plane is ax + by + cz = d. The normal n = u x (a,b,c)T and use A to find d.
Find two planes which intersect at a given point. Do they really ask this question? Just take any normals and substitute the point to find the two d.
Find a line that is perpendicular to a given line. Suppose that the given line is X1 = tu + A and the line you're looking for is X2 = sv + B. u x v = 0 solve for v. Use A for B.
Dernière édition par spirit le Dim 8 Mai 2011  22:06, édité 1 fois 

 
JellyFish
Messages : 166 Date d'inscription : 18/09/2007 Age : 23
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  20:15  
 It's too long to read, and so confusing! _________________ 

 
spirit
Messages : 222 Date d'inscription : 25/09/2007 Age : 24
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  20:20  
 Don't tell me I wasted my time for nothing. 

 
JellyFish
Messages : 166 Date d'inscription : 18/09/2007 Age : 23
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  20:21  
 _________________ 

 
Dark
Messages : 184 Date d'inscription : 13/05/2008
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  20:23  
 I have trouble with the adjoint. It's so long to calculate and so useless (well, so far...) Do you have any tips? _________________ 

 
spirit
Messages : 222 Date d'inscription : 25/09/2007 Age : 24
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  20:30  
 You can calculate the adjoint of A by using adjA = (detA)A^1, so you find the inverse of A and them find the adjoint. But usually, you use the adjoint to find the inverse of A, I think. 

 
Dark
Messages : 184 Date d'inscription : 13/05/2008
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  20:58  
 Ok. Other question: Use the properties of the dot product to show that u+v^2=u^2+v^2+2u*v _________________ 

 
spirit
Messages : 222 Date d'inscription : 25/09/2007 Age : 24
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  21:00  
 That's easy. u+v^2 = (u + v)*(u + v) = u*u + v*v + u*v + v*u = u^2 + v^2 + 2u*v 

 
Dark
Messages : 184 Date d'inscription : 13/05/2008
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  21:01  
 State and prove Lagrange’s Identity. _________________ 

 
spirit
Messages : 222 Date d'inscription : 25/09/2007 Age : 24
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  21:18  
 u x v^2 = u^2 v^2  (u*v)^2 = u^2 v^2  u^2 v^2 (cosθ)^2 = u^2 v^2 (1  (cosθ)^2) = u^2 v^2 (sinθ)^2 =u x v^2 QED 

 
Dark
Messages : 184 Date d'inscription : 13/05/2008
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  21:21  
 State the formula for the volume of the parallelepiped determined by the vectors u, v and w. Prove it. _________________ 

 
spirit
Messages : 222 Date d'inscription : 25/09/2007 Age : 24
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  21:32  
 It's the triple scalar product: u*(v x w) Why? Because  two vectors form a parallelogram: let them be v and w. The area of a parallelogram is base x height: let the base be v. The height would be w sinθ, thus area = vw sinθ = v x w. The volume of the parallelepiped is the area of the base x height. The height is lying on the vector v x w, thus it is proj_(v x w) u = u*(v x w)/v x w. Thus, the volume is v x w u*(v x w)/v x w = u*(v x w) 

 
Dark
Messages : 184 Date d'inscription : 13/05/2008
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  21:33  
 For some vector u and v, prove that u x v is orthogonal to both u and v. _________________ 

 
spirit
Messages : 222 Date d'inscription : 25/09/2007 Age : 24
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  21:53  
 u*u x v = v*u x u = v*0 = 0 Thus, they are orthogonal. Same for v. 

 
Dark
Messages : 184 Date d'inscription : 13/05/2008
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  21:55  
 Derive the formulas for plane and line equations in R3. _________________ 

 
spirit
Messages : 222 Date d'inscription : 25/09/2007 Age : 24
 Sujet: Re: Linear algebra study realm Dim 8 Mai 2011  22:13  
 Plane: A plane is defined by a point A(a1,a2,a3) and a normal n=(a,b,c)T. Taking any other point P(x,y,z), AP*n = 0. Thus, AP*n = (x  a1, y  a2, z  a3)T * (a,b,c) = a(x  a1) + b(y  a2) + c(z  a3) = ax  aa1 + by  ba2 + cz  ca3 = 0 ax + by + cz = aa1 + ba2 + ca3 = d The equation of a plane is ax + by + cz = d
Line: A line need two points P(p1,p2,p3) and Q(q1,q2,q3). PQ = (q1  p1, q2  p2, q3  p3)T The equation of the line is X = t(q1  p1, q2  p2, q3  p3)T + P or X = t(q1  p1, q2  p2, q3  p3)T + Q 

 
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