Homework Help: Basic geometry - dot product/cart. lines

1. Dec 16, 2012

boings

1. The problem statement, all variables and given/known data
We consider two points, B and I and a line 'a'.
B(0,-4,-7) I(-2,-2,-5) and a: x = y+1 = (z-2)/2

Determine the summits of A and C of triangle ABC knowing that:

-Summit A belongs to the line 'a'
-I is the foot of the height from A (perpendicular to BC)
-The angle of A is equal to arccos(1/3)

2. Relevant equations

dot product

3. The attempt at a solution

I feel as if I'm really close, but keep getting the wrong answer. My process is as follows: The dot product of BI and AI is equal to zero. Knowing this, I can obtain an equation of the line AI and determine where it intersects with line 'a'. This is as far as I've gotten in my attempt to find A.

2. Dec 16, 2012

I like Serena

Welcome to PF, boings!

Can you set up a parametric equation for the line 'a'?
That is, find a support vector and a direction vector?

If you fill that in for A in your dot product, you'll get an equation from which you can find A....

3. Dec 16, 2012

boings

Hi and thanks!

good point!

(if k=constant)
x= 0 + k
y= -1 + k
z= 2 + 2k

Although, when I plug that in it doesn't make much sense. I proceed like this:

AI (dot) BI = (-2, 2, 2)(dot)(-2, -2, -5)(0 + k, -1 + k, 2 + 2k)

I think that this is already flawed in some sense

4. Dec 16, 2012

I like Serena

Well, the vector AI is the difference of (-2, -2, -5) and (0 + k, -1 + k, 2 + 2k).
So that is (-2 - k, -1- k, -7 - 2k).
Then take the dot product and set it equal to zero...

5. Dec 16, 2012

boings

Alrighty, so I then get (4-2k, -2-2k, -14 - 4k)=0 which represents the line AI.

This is where I get a little tripped up. Should I solve for k and replace into original equation of the line? The answer should be: A(-3, -4, -4)

6. Dec 16, 2012

I like Serena

Hmm, (4-2k, -2-2k, -14 - 4k)=0 does not represent a line.
How come you think that?

The vector AI is represented by (-2 - k, -1- k, -7 - 2k) for some value of k.
The vector BI is (-2, 2, 2).
Their dot product has to be zero.

The dot product is (-2-k) x -2 + (-1-k) x 2 + (-7-2k) x 2 = 0.
Solve for k?

7. Dec 17, 2012

boings

Oh ok I get it! thank you so much.

I see how that's not a line now, but rather the equation that relies the orthogonality between AI and BI. It sure looks like a a parametric equation of a line though ^^

thanks again