Given the points $A(2,0,1)$, $B(0,1,1)$ and $C(0,3,2)$ in a coordinate system with positive orientation $(i,j,k)$. The problem asks to: a) Find the points $A$, $B$, and $C$. b) Find the coordinates of vectors $\vec{BA}$, $\vec{BC}$ and $\vec{AC}$. c) Calculate $\vec{BA} \cdot \vec{BC}$ and determine if triangle $ABC$ is a right triangle. d) Find the general equation of the sphere $(S)$ with center $A$ that passes through $B$. e) Additional question related to the circle described by the vector $(AB)$

1. Problem Description

Given the points $A(2,0,1)$, $B(0,1,1)$ and $C(0,3,2)$ in a coordinate system with positive orientation $(i,j,k)$. The problem asks to:

a) Find the points $A$, $B$, and $C$.

b) Find the coordinates of vectors $\vec{BA}$, $\vec{BC}$ and $\vec{AC}$.

c) Calculate $\vec{BA} \cdot \vec{BC}$ and determine if triangle $ABC$ is a right triangle.

d) Find the general equation of the sphere $(S)$ with center $A$ that passes through $B$.

e) Additional question related to the circle described by the vector $(AB)$

2. Solution Steps

a) The points are given as $A(2,0,1)$, $B(0,1,1)$, and $C(0,3,2)$.

b) Find the coordinates of vectors $\vec{BA}$, $\vec{BC}$ and $\vec{AC}$.

$\vec{BA} = A - B = (2,0,1) - (0,1,1) = (2, -1, 0)$

$\vec{BC} = C - B = (0,3,2) - (0,1,1) = (0, 2, 1)$

$\vec{AC} = C - A = (0,3,2) - (2,0,1) = (-2, 3, 1)$

c) Calculate $\vec{BA} \cdot \vec{BC}$ and determine if triangle $ABC$ is a right triangle.

$\vec{BA} \cdot \vec{BC} = (2, -1, 0) \cdot (0, 2, 1) = (2)(0) + (-1)(2) + (0)(1) = 0 - 2 + 0 = -2$

Since $\vec{BA} \cdot \vec{BC} \neq 0$, the vectors $\vec{BA}$ and $\vec{BC}$ are not orthogonal. So, the angle at vertex $B$ is not a right angle.

Let's check $\vec{BA} \cdot \vec{AC} = (2,-1,0) \cdot (-2,3,1) = (2)(-2) + (-1)(3) + (0)(1) = -4 - 3 + 0 = -7 \neq 0$. So the angle at $A$ is not a right angle.

Let's check $\vec{AC} \cdot \vec{BC} = (-2,3,1) \cdot (0,2,1) = (-2)(0) + (3)(2) + (1)(1) = 0 + 6 + 1 = 7 \neq 0$. So the angle at $C$ is not a right angle.

Therefore, triangle $ABC$ is not a right triangle.

d) Find the general equation of the sphere $(S)$ with center $A$ that passes through $B$.

The center of the sphere is $A(2,0,1)$.

The radius of the sphere is the distance between $A$ and $B$.

$r = AB = \sqrt{(2-0)^2 + (0-1)^2 + (1-1)^2} = \sqrt{2^2 + (-1)^2 + 0^2} = \sqrt{4 + 1 + 0} = \sqrt{5}$

The equation of the sphere with center $(x_0, y_0, z_0)$ and radius $r$ is $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2$.

In this case, the equation of the sphere is $(x - 2)^2 + (y - 0)^2 + (z - 1)^2 = (\sqrt{5})^2$.

$(x - 2)^2 + y^2 + (z - 1)^2 = 5$

Expanding this gives:

$x^2 - 4x + 4 + y^2 + z^2 - 2z + 1 = 5$

$x^2 + y^2 + z^2 - 4x - 2z + 5 = 5$

$x^2 + y^2 + z^2 - 4x - 2z = 0$

e) Additional question related to the circle described by the vector $(AB)$. This part requires further clarification of the question. The circle does not relate to a vector $(AB)$. Vector is a line from $A$ to $B$. A circle requires a center and a radius, both of which are not indicated clearly in this prompt.

3. Final Answer

a) $A(2,0,1)$, $B(0,1,1)$, $C(0,3,2)$

b) $\vec{BA} = (2, -1, 0)$, $\vec{BC} = (0, 2, 1)$, $\vec{AC} = (-2, 3, 1)$

c) $\vec{BA} \cdot \vec{BC} = -2$. Triangle $ABC$ is not a right triangle.

d) $x^2 + y^2 + z^2 - 4x - 2z = 0$

e) Insufficient information is provided.