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Find scalars a, b, c, d, e, f, and g so that the matrix A is orthogonal.
 one year ago
 one year ago
Find scalars a, b, c, d, e, f, and g so that the matrix A is orthogonal.
 one year ago
 one year ago

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brinetheryBest ResponseYou've already chosen the best response.1
It would be really nice to get an explanation of how to go about doing this problem :)
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
There are two properties that orthogonal matrices have. When looking at the columns of the matrix, in this case the vectors v1, v2 and v3, if you take the inner product (or dot product) of a column with itself, you should get 1, and if you take the inner (dot) product of a column with any other column, you should get 0. Some how we need to use these two properties to figure out what the correct numbers are.
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
Using the first property, we can conclude that b = e = 0, since:\[v_2\cdot v_2 = 1 \Longrightarrow b^2 +e^2 +1^2 =1\] which only has the solution b = 0 and e = 0. Does that make sense?
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
For the first property, you're saying the the magnitude should just be 1?
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
yes that is correct, the magnitude of each column should be 1.
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
I'm rereading your paragraph, one sec :)
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
sure thing, sry for rushing.
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
It's no rush, I am just a slow learner lol
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
So it looks like "a" has to equal a number and "d" also has to equal a number (both nonzero, of course)
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
Does my explanation of why b = 0 and e = 0 make sense?
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
sorry, OS froze up.
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
Good. Now maybe we can use the second property to figure out some stuff.
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
Maybe taking the dot product of the second column with any of the other two columns will shed some light on this?
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
just a sec, let me go grab the white board
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
just one step at a time. Remember, we now know the entire second column, from top to bottom its (0, 0, 1).
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
So I wrote it up and it looks like g= 0. And then to find what the entire third column is, just take the magnitude. so sqrt(c^2 +(3/5)^2 +0^2)?
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
that is correct :)
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
haha nevermind again! I got confused for a sec with what I was doing. Okay, so I took the magnitude and set it equal to 1. I got c=sqrt(16/25), or 4/5
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
ok, so now we know the second and third columns completely. Lets see what damage can be done to the first column,
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
I tried doing the dot product but that doesn't look like it's getting me anywhere.
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
what happens when you dot the second and the first columns? Think similarly to how you got that g = 0, but look at f this time.
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
Oh, I see what you're saying. I tried dotting all of the columns. hehe
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
Okay, so f=0, but either a or d must equal 1.
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
or some other number for both a and d?
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
hmm...from the first property, we have that:\[v_1\cdot v_1=1\Longrightarrow a^2 +d^2+0^2=1\Longrightarrow a^2+d^2=1\]and from taking the dot product of the first and tthird column, property two tells us:\[v_1\cdot v_3 = 0\Longrightarrow \frac{4}{5}a+\frac{3}{5}d+0\cdot 0=0\Longrightarrow \frac{4}{5}a+\frac{3}{5}d=0\]Maybe we can use these equations together as a system of equations to solve for a and d?
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
Oh my gosh, I did not even see that.
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
Thank you very much for writing all of that up.
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
So a= sqrt(1d^2) and sub that into the second eqn and then solve for d?
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
While that will work, it might be a little easier to say:\[\frac{4}{5}a+\frac{3}{5}d=0\iff \frac{4}{5}a=\frac{3}{5}d\iff a=\frac{3}{4}d\]and substitute that into the first equation. Again, its not what you said is wrong, Im just trying to avoid square roots since they make things a little complicated.
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
Thank you for taking the time to really explain this to me. I can barely decipher what is written in my linear algebra book.
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
which book are you using?
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
linear algebra with applications by Otto Bretscher
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
If you need more resources, other than your book, MIT's OCW videos are decent. Here's their video on orthogonal matries: http://ocw.mit.edu/courses/mathematics/1806linearalgebraspring2010/videolectures/lecture17orthogonalmatricesandgramschmidt/
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
I am lost on another question as well. Would you have time to help me with one more? That will be all, I promise :)
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
If not, that's okay. I totally understand and you've helped me so much already.
 one year ago

joemath314159Best ResponseYou've already chosen the best response.1
My time is a little short, but what is the question?
 one year ago

brinetheryBest ResponseYou've already chosen the best response.1
I'll just make a new post and give you the link. If you're busy, I understand completely.
 one year ago

MathmuseBest ResponseYou've already chosen the best response.0
Thanks @Joemath314159 I've learned something. Just to add now that @brinethery has got it, since the transpose of a orthogonal matrix is also orthogonal, you can perform the same operations with dot product on the row vectors no?
 one year ago
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