Finding the angle between surfaces

[asd1249jf]

Homework Statement

http://img526.imageshack.us/img526/4890/asdny3.jpg

*Length of each side of the square is 3.
*Length of DP = Length of BQ = Length of GR = 1

Calculate $$\theta$$, the angle that is made between the surface of PQR and the surface of CGHD

Homework Equations

No relevant equation was given for me, but I'm going to throw out some that may help.

*Law of Cosines
*Dot Product

The Attempt at a Solution

I do not have a single clue of what to do. Can anyone start me off somehow, and tell me what I should be initially looking for in order to find the angle between the surfaces?

 

[robphy]

The angle between the surfaces is equal to the angle between the normal-vectors to these surfaces... which you can determine using the dot-product.
You have enough information to determine each normal-vector.

 

[asd1249jf]

Wait, but I thought normal vector was found using the cross product, not the dot product?

 

[robphy]

Wait, but I thought normal vector was found using the cross product, not the dot product?

That's one way to do it (and that's the best way for your problem).
Now, once you found those normal vectors using the cross-product,
now use the dot product to find the angle between those [normal] vectors.

 

[asd1249jf]

Well I think I can find the normal vector of the flat surface, but how would you find the normal vector of the triangular surface?

 

[robphy]

Well I think I can find the normal vector of the flat surface, but how would you find the normal vector of the triangular surface?

Can you find two linearly-independent [i.e. two non-parallel] vectors on that triangular surface?

 

[asd1249jf]

Pq = (3,-1,0) Pr = (3,0,-3)

 

[asd1249jf]

where do I go from here

 

[robphy]

Use the cross-product, as you said in #3.

 

[asd1249jf]

? I don't understand how that would help me, as that would produce a vector that is normal to the two vectors given above, NOT the normal vector that is perpendicular to the surface.

I'll do it anyways.

(3,9,3)

 

[HallsofIvy]

If two vectors lie IN a surface, then the normal to the surface must be normal to both. In fact, unless the two vectors lie along the same line (in which case their cross product would be 0) their cross product IS normal to the surface.

 

[asd1249jf]

Ok, if that is the case, let me try to solve this

Dc = (3,0,0) Dh = (0,0,-3)

Dc X Dh = (0,9,0) = A

Pq X Pr = (3,9,3) = B

$$A * B = |A||B|\cos(\theta)$$

$$\cos^{-1}\frac{A * B}{|A||B|} = \theta$$

$$\theta = \cos^{-1}\frac{81}{|9||9.798|}$$

$$\theta = 23.284288 degrees$$

...And that is an incorrect answer, where did I go wrong?

 

[HallsofIvy]

The FIRST thing you had better do is set up a coordinate system so that vectors from one point to another make sense! Set up a coordinate system with origin at F, x-axis along FG, y-axis along FE, z-axis along FB. Then the vector perpendicular to CGHD is <1,0, 0> and the vector perpendicular to plane PQR is the cross product of QP= <1,0,1>-<2,1,1>= <-1, -1, 0> and QR= <1,0,1>- <2,0,0>= <-1,0,1>. That cross product is <-1,1,-1>. The angle is given by <1, 0, 0>.<-1, 1, -1>= |<1,0,0>||<1,1,1>|cos($\theta$ or -1= (1)($\sqrt{3}$)cos($\theta$)which gives cos($\theta)= -1/\sqrt{3}$. What is $\theta$?

 

[asd1249jf]

$$\theta = 125.26439 degrees$$

And according to the answer, it is still wrong.

The answer is supposed to be

$$\cos(\theta) = \frac{3\sqrt{11}}{11}$$

 

[asd1249jf]

I can't find anything you did wrong either. Would the answer be wrong?

 

[msslowlearner]

the normals as you calculated were right .. maybe the simplification of the cos theta part may have gone wrong ...

i proceeded as hallsofvy said in #13, but i got cos theta = -3/sqrt 11

 

[Char. Limit]

the normals as you calculated were right .. maybe the simplification of the cos theta part may have gone wrong ...

i proceeded as hallsofvy said in #13, but i got cos theta = -3/sqrt 11

Remember that the usual range of the arccosine function is [0,pi]. So you'll need to find a POSITIVE value of theta. Also, remember that cos(x)=cos(-x).

 

[msslowlearner]

yes, but here my value of theta is 154.75 degrees when it should be 25.24 degrees. the only difference being the sign of theta. The magnitude of cos theta is right while the direction is reversed .. where could i have gone wrong ?

 

[ehild]

When two straight line cross, there are two angles between them: one smaller than 90°, the other greater than 90°, and the sum of those two angles is 180°. The angle of two straight lines is defined by the smaller one.

ehild