Given the points A(1, 2, 3), B(5, 4, 3), and C(2, 1, 2), find angle ABC.

We have to find the angle ABC given the points A( 1,2,3),
B(5,4,3) and C(2,1,2)

This can be done by using the law of
cosines.

First we need to determine the length of AB, BC
and CA.

The length of AB = sqrt[( 5-1)^2 +(4 -2)^2 +(3 -
3)^2] = sqrt( 16 + 4) = sqrt 20.

Similarly BC = sqrt[(
5-2)^2 +(4 -1)^2 +(3 - 2)^2] = sqrt( 9 + 9 + 1) = sqrt
19.

And CA = sqrt[( 2-1)^2 +(1 -2)^2 +(3 - 2)^2] = sqrt( 1
+ 1 + 1) = sqrt 3.

According to the law of cosines, we have
CA^2 = AB^2 + BC^2 - 2*AB*BC*cos ABC

Substituting the
values we have obtained:

3 = 20 + 19 - 2*sqrt 19 * sqrt 20
* cos ABC

=> cos ABC = (20 + 19 - 3)/( 2* sqrt 20 *
sqrt 19)

=> cos ABC = 36/( 2* sqrt 20 * sqrt
19)

=> cos ABC = 18/(sqrt 20 * sqrt
19)

=> ABC = arc cos [18/(sqrt 20 * sqrt
19)]

=> ABC = 22.57
degrees.

The required angle ABC is 22.57
degrees.