Vector triangle question - please check me on this

[bcjochim07]

Homework Statement

The vertices of a triangle are given by points A: (2,-7,3) B: (-1,5,8), & C: (4,6,-1) Is this triangle acute, obtuse, or right?

Homework Equations

dot product is positive : acute angle
dot product is negative : obtuse angle

The Attempt at a Solution

My main question is: don't the vectors have to be arranged tail to tail before you can take the dot product to determine the angle between the vectors? That seems to be the way it is defined in my book.

The three vectors that make up the triangle are
AB: [-3,12,5]
AC: [2,13,-4]
BC: [5,1,-9]

AB\cdotAC= (-3)(2) + (12)(13) + (5)(-4)= 130 > 0, so angle BAC is acute.

AC\cdotBC = (2)(5) + (13)(1) + (-4)(-9) = 59 > 0, so angle ACB is acute
(This is the same as CA\cdotCB so they are tail to tail)

BC\cdotAB = (5)(-3) + (1)(12) + (-9)(5) = -48 < 0, but this is not the angle in the triangle, according to the picture I drew. Rather, this is the supplement, so the angle ABC is also acute.

Thus, the triangle is acute. Could somebody check my work please? Thanks.

 

[bcjochim07]

nota bene: The ABAC, ACBC, & BCAB are supposed to be dot products.

 

[bcjochim07]

Any ideas? This assignment is due tomorrow, and I'm pretty curious about whether I am doing this problem correctly or not.