The following problems will be used to study for a quiz from, since I
have already completed most of these problems, no explanations in
words are necessary, but I will need to see step-by-step how each
problem has been solved. Also, I know that $3.00 is not a huge amount
of money for these problems, but they are simple and should only take
a few minutes each, and I will be putting up many more difficult(more
expensive?) problems as the year progesses, and I will keep in mind
who provides good answers that are helpful and who answers
quickly.(P.S. calebu2-ga has been a great researcher thus far):)
1. A^-1=[3,2;1,3] b^-1=[2,5;3,-2]
Find A and B
2. Find the area of the right triangle with vertices (0, 0),
(0,3),(4,0). Verify by using the formula A=1/2(base)(height)
3. Which of the vectors u1=(1,2), u2=(0,1),
u3=(-2,-4),u4=(-2,1),u5=(2,4),u6=(-6,3) are:
(a)Orthogonal?
(b)In the same direction?
(c)In opposite directions?
4. Show that if w is orthogonal to u and v, then w is orthogonal to
ru+sv, where r and s are the scalars.
5. For questions 2, 3, and 4: In addition to solving the given
problems, please provide a similar problem to the one given, except 3
dimensional, and provide the respective answer.
Thanks for all of your time,
Bildy-ga Rutgers Preparatory School - Courses & Curriculum::
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Philosophy of Teaching Mathematics::
etc. This is why it makes sense to ask simple questions such as so (I am writing 2. For some people the time-constrained atmosphere of a quiz is a problem
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If I am not mistaken, the general equation for finding an area of triangle is
A = 1/2 v x w
where v and w are vectors defining the triangle. These work in 3 space too, I think.
Forgive me for being na ve, but I don't recollect, and my search did
not yield, any formula for finding the area of a triangle in 3-D space
using linear algebra.
Would you enlighten me about the formula that you learned? (I know the
2-D formula, but not the 3-D one.)
Thanks
secret901
If you can complete everything except for the 3D area of a triangle
question then please do so The Math Forum - Math Library - Basic Algebra::
Library is a comprehensive catalog of Web sites and Web pages relating to the study of mathematics. This page contains sites relating to Basic Algebra.
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Oh what the heck... let me just do it. Errors and Omissions Excepted.
1) A = 1/7 [3,-2;-1,3]
B = 1/-19 [-2,-5;-3,2]
2) A = 1/2 bh = 1/2 * 4 * 3 = 6
A = 1/2 0 0 4 0
0 3 0 0
= 1/2 ( 0 - 0 + 0 - 12 + 0 - 0 )
= -6 (signed) so the unsigned area is 6
3) there are 6 vectors to test, but by inspection, I managed to pick out:
a) u1.u4 = 0
b) u1 and u5
c) u1 and u3, u3 and u5
4) as shown above
5) Heh heh... A-1 = I (3x3) B-1 = I-1 (3x3)
therefore A = I, B = I. No, seriously, use the other relationship.
for a 3D triangle, A(1,0,0) B(0,1,0) C(0,0,1), the area is given by
let v = B - A = <-1,1,0>
w = C - A = <-1,0,1>
v x w = <1,1,1>
A = 1/2 v x w = 1/2 sqrt( 1 + 1 + 1) = sqrt(3)/2
I'm not sure if this absolutely right though... someone please confirm this.
Just for the heck of it, I'm going to outline some principles...
numerical answers can easily be gotten using Mathematica or Maple.
1. for 2-space, matrix inversion is done using:
where A = [a,b;c,d]
the inverse A-1 = 1/(ad-bc)[a,-b;-c,a]
in 3-space, A-1 = 1/A adj (A)T.
That's the reciprocal of the determinant of A multiplied by adjunct
matrix of the transposed A.
2. Use Cramer's rule:
A = 1/2 x1 x2 x3 x1
y1 y2 y3 y1
and verify with
A = 1/2 b h
using a transformation. Or use A = 1/2 v x w (the x is a cross
product, not a multiplication)
3. The test for orthogonality is always the dot product. Just dot the
pairs and see if they're zero.. if yes, then they're orthogonal. As
for the others, it's a matter of simple vector algebra. if a vector A
is parallel to a vector B, then A = kB where k is a constant. if k is
negative, then the vectors are in opposite directions.
4. Using vector rules
w.u = 0
w.v = 0
to show that w.(ru + sv) = 0
L.H.S. = w.(ru + sv)
= w.(ru) + w.(sv)
= r w.u + s w.v
= 0 + 0
= R.H.S.
Disclaimer: I haven't done linear algebra for years, so I might be
mistaken about some things Free Statistical Software::
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Math::
Intermediate Algebra moves beyond linear and quadratic functions to study The objective of the course is to prepare the student for AP Calculus AB, a
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I was told everything was done the same in 2D as it is done in 3D
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Some Simple Linear Algebra Question To Prepare For A Quiz