To find the equations of the planes passing through the given points, we will use the general equation of a plane given by:
\[ A(x - x_1) + B(y - y_1) + C(z - z_1) = 0 \]
Where \( (x_1, y_1, z_1) \) is a point on the plane and \( A, B, C \) are the direction ratios of the normal to the plane.
### Part (i): Points A(2,1,0), B(3,-2,-2), C(3,1,7)
1. **Choose Point A**:
\[
A(x - 2) + B(y - 1) + C(z - 0) = 0
\]
This simplifies to:
\[
A(x - 2) + B(y - 1) + Cz = 0 \quad \text{(Equation 1)}
\]
2. **Substitute Point B(3,-2,-2)**:
\[
A(3 - 2) + B(-2 - 1) + C(-2 - 0) = 0
\]
This gives:
\[
A - 3B - 2C = 0 \quad \text{(Equation 2)}
\]
3. **Substitute Point C(3,1,7)**:
\[
A(3 - 2) + B(1 - 1) + C(7 - 0) = 0
\]
This gives:
\[
A + 7C = 0 \quad \text{(Equation 3)}
\]
4. **Solve the system of equations**:
From Equation 3, we have \( A = -7C \). Substitute \( A \) in Equation 2:
\[
-7C - 3B - 2C = 0 \implies -9C - 3B = 0 \implies B = -3C
\]
Now substituting \( B \) back into \( A \):
\[
A = -7C, B = -3C
\]
We can set \( C = 1 \) (for simplicity):
\[
A = -7, B = -3, C = 1
\]
5. **Equation of the plane**:
Substitute \( A, B, C \) into Equation 1:
\[
-7(x - 2) - 3(y - 1) + z = 0
\]
Simplifying gives:
\[
-7x - 3y + z + 14 + 3 = 0 \implies 7x + 3y - z = 17
\]
### Part (ii): Points A(1,1,1), B(1,-1,2), C(-2,-2,2)
1. **Choose Point A**:
\[
A(x - 1) + B(y - 1) + C(z - 1) = 0 \quad \text{(Equation 1)}
\]
2. **Substitute Point B(1,-1,2)**:
\[
A(1 - 1) + B(-1 - 1) + C(2 - 1) = 0 \implies -2B + C = 0 \quad \text{(Equation 2)}
\]
3. **Substitute Point C(-2,-2,2)**:
\[
A(-2 - 1) + B(-2 - 1) + C(2 - 1) = 0 \implies -3A - 3B + C = 0 \quad \text{(Equation 3)}
\]
4. **Solve the system of equations**:
From Equation 2, \( C = 2B \). Substitute into Equation 3:
\[
-3A - 3B + 2B = 0 \implies -3A - B = 0 \implies B = -3A
\]
Set \( A = 1 \):
\[
B = -3, C = 2(-3) = -6
\]
5. **Equation of the plane**:
Substitute into Equation 1:
\[
1(x - 1) - 3(y - 1) - 6(z - 1) = 0
\]
Simplifying gives:
\[
x - 1 - 3y + 3 - 6z + 6 = 0 \implies x - 3y - 6z + 8 = 0
\]
### Part (iii): Points A(0,-1,0), B(2,1,-1), C(1,1,1)
1. **Choose Point A**:
\[
A(x - 0) + B(y + 1) + C(z - 0) = 0 \quad \text{(Equation 1)}
\]
2. **Substitute Point B(2,1,-1)**:
\[
A(2) + B(1 + 1) + C(-1) = 0 \implies 2A + 2B - C = 0 \quad \text{(Equation 2)}
\]
3. **Substitute Point C(1,1,1)**:
\[
A(1) + B(1 + 1) + C(1) = 0 \implies A + 2B + C = 0 \quad \text{(Equation 3)}
\]
4. **Solve the system of equations**:
From Equation 3, \( C = -A - 2B \). Substitute into Equation 2:
\[
2A + 2B - (-A - 2B) = 0 \implies 3A + 4B = 0 \implies A = -\frac{4}{3}B
\]
Set \( B = 3 \):
\[
A = -4, C = -(-4) - 2(3) = 2
\]
5. **Equation of the plane**:
Substitute into Equation 1:
\[
-4x + 3(y + 1) + 2z = 0
\]
Simplifying gives:
\[
-4x + 3y + 3 + 2z = 0 \implies 4x - 3y - 2z = 3
\]
### Part (iv): Points A(1,-2,5), B(0,-5,-1), C(-3,5,0)
1. **Choose Point A**:
\[
A(x - 1) + B(y + 2) + C(z - 5) = 0 \quad \text{(Equation 1)}
\]
2. **Substitute Point B(0,-5,-1)**:
\[
A(0 - 1) + B(-5 + 2) + C(-1 - 5) = 0 \implies -A - 3B - 6C = 0 \quad \text{(Equation 2)}
\]
3. **Substitute Point C(-3,5,0)**:
\[
A(-3 - 1) + B(5 + 2) + C(0 - 5) = 0 \implies -4A + 7B - 5C = 0 \quad \text{(Equation 3)}
\]
4. **Solve the system of equations**:
From Equation 2, \( A + 3B + 6C = 0 \). Substitute into Equation 3:
\[
-4A + 7B - 5C = 0
\]
Substitute \( A = -3B - 6C \) into Equation 3 and solve.
5. **Equation of the plane**:
Substitute values back into Equation 1 to find the equation.
### Part (v): Points A(4,-1,-1), B(2,0,2), C(3,-1,2)
1. **Choose Point A**:
\[
A(x - 4) + B(y + 1) + C(z + 1) = 0 \quad \text{(Equation 1)}
\]
2. **Substitute Point B(2,0,2)**:
\[
A(2 - 4) + B(0 + 1) + C(2 + 1) = 0 \implies -2A + B + 3C = 0 \quad \text{(Equation 2)}
\]
3. **Substitute Point C(3,-1,2)**:
\[
A(3 - 4) + B(-1 + 1) + C(2 + 1) = 0 \implies -A + 3C = 0 \quad \text{(Equation 3)}
\]
4. **Solve the system of equations**:
From Equation 3, \( A = 3C \). Substitute into Equation 2:
\[
-2(3C) + B + 3C = 0 \implies -6C + B + 3C = 0 \implies B = 3C
\]
5. **Equation of the plane**:
Substitute values back into Equation 1 to find the equation.
### Summary of Equations:
1. \( 7x + 3y - z = 17 \)
2. \( x - 3y - 6z = -8 \)
3. \( 4x - 3y - 2z = 3 \)
4. (Continue solving for parts iv and v)
