The angle between the intersection of a line and plane

• Mar 30th 2014, 02:54 PM
sakonpure6
The angle between the intersection of a line and plane
For question 20 a), The answer in the back of the book is 16 degrees, I am getting 22 degrees. My approach to the solution was to calculate the point of intersection which I got (4, -3/2 , 1). Then I let y=z=0 on the equation of the plane to find a point on it which in this case is (2,0,0). So, we have two points on the plane (2,0,0) and (4, -3/2 , 1) and so we can find another direction vector of which I calculated (2,-3/2 ,1).

Now using the formula $cos\theta = \frac{d1 (dot) d2 }{|d1| \times |d2|}$ I get 22 degrees.

Why am I wrong or why is my approach incorrect.

Thank you

Attachment 30563
Attachment 30564
• Mar 30th 2014, 03:41 PM
Plato
Re: The angle between the intersection of a line and plane
Quote:

Originally Posted by sakonpure6
For question 20 a), The answer in the back of the book is 16 degrees,]

I also get about $16^o$.

First there is absolutely no need to find the point of intersection.

The direction vector of the line is $\vec{D}=<2,-1,2>$. The normal to the plane is $\vec{N}=<1,2,1>$.

The angle between the line and the plane is $\dfrac{\pi}{2}-\arccos\left(\dfrac{|\vec{D}\cdot\vec{N}|}{\|\vec{ D}\|~\|\vec{N}\|}\right)$

See the calculation here.
• Mar 30th 2014, 03:47 PM
sakonpure6
Re: The angle between the intersection of a line and plane
Thank you for the reply, but I have not learned arccos yet. So how would you represent this answer using the normal trig ratios?
• Mar 30th 2014, 04:00 PM
Plato
Re: The angle between the intersection of a line and plane
Quote:

Originally Posted by sakonpure6
Thank you for the reply, but I have not learned arccos yet. So how would you represent this answer using the normal trig ratios?

What do you mean that "I have not learned arccos yet"?
Why are you given a problem that requires that function to solve?

Frankly I don't see how any competent education authority could offer vectors at this level without insuring that the students have the necessary mathematical grounding. In this case it certainly includes the $arccos$ function.
• Mar 30th 2014, 04:14 PM
sakonpure6
Re: The angle between the intersection of a line and plane
Well as a student, the Ontario ministry of education has let me down. Either way, can you spot why my approach is wrong?

Thank you though!
• Mar 30th 2014, 04:29 PM
Plato
Re: The angle between the intersection of a line and plane
Quote:

Originally Posted by sakonpure6
can you spot why my approach is wrong?

Frankly, I see nothing in your approach that has anything to do with this question. So I cannot answer that.

I taught vector geometry and vector analysis for over thirty years. In fact, I helped develop core standards for vector geometry here below your border.

But your are correct, I know absolutely nothing about the standards in Ontario. Maybe you can find a Canadian tutor.
• Mar 30th 2014, 04:35 PM
sakonpure6
Re: The angle between the intersection of a line and plane
I'm so sorry If I offended you, I had no such intentions :(
• Mar 30th 2014, 05:22 PM
Plato
Re: The angle between the intersection of a line and plane
Quote:

Originally Posted by sakonpure6
I'm so sorry If I offended you, I had no such intentions :(

Absolutely not. You gave no offence. Put that out of your brain.

It is true that I am offended by any system would fail students like you find yourself part of.

How have you found the angle between two vectors? It only can be done with the $\arccos$ function.