Expert: Scotto - 9/25/2009

Question
Hi Scotto the question is

Use dot product to find the lenghts of the sides of the triangle with Vertices at positions (0,0,0), (1,2,3) and (4,-1,3). And then find the three internal angles in the triangle and conferm that they add to 180Deg


Thanks
Matt

Answer
You might here this from other assistance, but the private should be, 'Yes'.
Since it is a no, you'll get responses from other assistants as well.

To find the lengths, compute the distance vector and then use the dot product to get the length².
That is, from (0,0,0) to (1,2,3) is (1,2,3).  The dot product of this with itself is P=1²+2²+3².
The length L = √P, so the length L is √14.

For (4,-1,3) - (0,0,0), the vector is (4,-1,3).
The dot product is 4²+(-1)²+3² = 16 + 1 + 9 = 26, so the length L is √P = √26.

For the third one, the directional vector is (1,2,3) - (4,-1,3) = (-3,3,0).
The dot product of this with itself is 9+9+0 = 18.  L = √18 = 2√3.

To find each angle, use the cosine law.  This says a² + b² - 2ab•cos(C) = c².
To find the angle, note that this equation can be rearranged as
2ab•cos(c) = a² + b² - c², so cos(c) = (a² + b² - c²)/(2ab), so c = arccos((a² + b² - c²)/(2ab)).

Using the three sides, √14, √26, and 2√3, and noting that the squares are
14, 26, and 12, respectively, the three angles can be found.
For the 1st, let a=√14, b=√26, and c=2√3.
For the 2nd, let a=√26, b=2√3, and c=√14.
For the 3rd, let a=2√3, b=√14, and c=√26.