Linear Algebra Question, tilting plane angle?

[Canadian]

[SOLVED] Linear Algebra Question, tilting plane angle?

Homework Statement

A million years ago, an alien species built a vertical tower on a horizontal plane. When they returned they discovered that the ground had tilted so that measurements of 3 points on the ground gave coordinates of (0, 0, 0), (3, 1, 0), and (0, 2, 1). By what angle does the tower now deviate from the vertical?

Homework Equations

cross product, dot product.

The Attempt at a Solution

I'm pretty sure that I have the theory, what I'm not sure on is where the x, y and z axis are in relation to "horizontal" and "vertical". I'm thinking that to find the answer I need to use the normals of the planes (found by cross product of two vectors derived from points). I would then use a modified dot product formula to get the angle.

The stumbling block I'm running into is assigning points (and therefor vectors and a plane) to the "horizontal plane" in the question, which (x, y, or z) is pointing "up". If I take z as the upward axis I get,

Original plane: normal (0,0,1)
Tilted plane: points, (0,0,0), (3,1,0), and (0,2,1), to get vectors (3,1,0), and (0,2,1). cross product to get normal of (1,-3, 6),

use dot product to get that the angle must be (cos-1)(6/(sqrt(46))

Am I on the right track??

Edited to add, I am having trouble with some linear transformation problems too so have added them too. Thanks! one has an attempted solution (wrong) the other I am completely stuck on but have the theory on the bottom right.

Homework Statement

I'm not sure how to take the theory we learned and apply it to this question.

Homework Statement

Attempted solution, is wrong. The original question reads.

 

[HallsofIvy]

Homework Statement

A million years ago, an alien species built a vertical tower on a horizontal plane. When they returned they discovered that the ground had tilted so that measurements of 3 points on the ground gave coordinates of (0, 0, 0), (3, 1, 0), and (0, 2, 1). By what angle does the tower now deviate from the vertical?

Homework Equations

cross product, dot product.

The Attempt at a Solution

I'm pretty sure that I have the theory, what I'm not sure on is where the x, y and z axis are in relation to "horizontal" and "vertical". I'm thinking that to find the answer I need to use the normals of the planes (found by cross product of two vectors derived from points). I would then use a modified dot product formula to get the angle.

The stumbling block I'm running into is assigning points (and therefor vectors and a plane) to the "horizontal plane" in the question, which (x, y, or z) is pointing "up". If I take z as the upward axis I get,

Original plane: normal (0,0,1)
Tilted plane: points, (0,0,0), (3,1,0), and (0,2,1), to get vectors (3,1,0), and (0,2,1). cross product to get normal of (1,-3, 6),

use dot product to get that the angle must be (cos-1)(6/(sqrt(46))

Am I on the right track??

Yes, that's exactly the way I would do it.

Edited to add, I am having trouble with some linear transformation problems too so have added them too. Thanks! one has an attempted solution (wrong) the other I am completely stuck on but have the theory on the bottom right.

Which problem is which? The first one gives a number of functions and asks you to tell which are linear transformations: any linear transformation on a finite dimensional space (all these spaces are finite dimensional) can be written as a matrix multiplication. In particular, that means that each component of the result is a sum of numbers times the components of the orginal vector. It should be easy to see which are.

Homework Statement

I'm not sure how to take the theory we learned and apply it to this question.

Homework Statement

Attempted solution, is wrong. The original question reads.

[/QUOTE]

In your last problem, it's not clear to me what you intend "c1" and "c2" to be!

What I would do is write [x, y] as a linear combination of [-1, 2] and [1, -3]. That is, find c1, c2 such that x= -c1+ c2, y= 2c1- 3c2. Of course they will depend on x and y.
Since you are told That T[-1,2]= [17, -19] and that T[1, -3]= [-24, 24], you have T[x,y]= c1T[-1,2]+ c2T[1, -3]= c1[17, -19]+ c2[-24, 24].

 

[Canadian]

Alien question is solved, thank you for your help.

The first one gives a number of functions and asks you to tell which are linear transformations: any linear transformation on a finite dimensional space (all these spaces are finite dimensional) can be written as a matrix multiplication. In particular, that means that each component of the result is a sum of numbers times the components of the orginal vector. It should be easy to see which are.

I still am not getting this question, I think I am making it more complicated than it needs to be. Would you be able to give me an example of how this works (using one from the question or another)?

In your last problem, it's not clear to me what you intend "c1" and "c2" to be!

What I would do is write [x, y] as a linear combination of [-1, 2] and [1, -3]. That is, find c1, c2 such that x= -c1+ c2, y= 2c1- 3c2. Of course they will depend on x and y.
Since you are told That T[-1,2]= [17, -19] and that T[1, -3]= [-24, 24], you have T[x,y]= c1T[-1,2]+ c2T[1, -3]= c1[17, -19]+ c2[-24, 24].

Thanks, I am really not sure myself, I was basing my work off of a problem we worked in class. I was trying to solve the problem by finding the standard matrix then transforming the arbitrary point (x, y) using that matrix. I was reading the question as it asking what the vector (x,y) would be after performing the transformation on it. I am still confused by this question.

If I do this my standard matrix comes back to be,

[ -3 7]
[ 9 -5]

Then (x,y) will be (6x, 2y) but this is wrong.

My goodness this shouldn't be this hard!

 

[HallsofIvy]

To take an easy one, if y1= 3x1+ 2x2, y2= 4, y3= 5x1+ 2x2+ 3x3, I would decide immediately that this is NOT a linear transfromation by looking at the formula for y2: If x1, x2, x3 were doubled y2 would not change at all. Yet, from T(2v)= 2T(v), it would have to double as well.

Another easy one: If y1= x1*x2, y2 and y3 could be anything by I already know this is not LINEAR! In particular, if 2v= (2x1, 2x2, 2x3) then y1 changes to (2x1)(2x2)= 4x1*x2, not twice.

ANY linear transformation must be of the form y1= a1x1+ a2x2+a3x3, y2= b1x1+ b2x2+ b3x3, y3= c1x1+ c2x2+ c3x3: sums of numbers times the components. Anything else is not a linear transformation and cannot satisfy T(av)= aT(v), T(u+ v)= T(u)+ T(v).

 

[Canadian]

Great thanks!