Using section formula, show that the points A(1, 2, 1), B(2, 1, 0) and `c(7/5, 8/5, 3/5)` are
collinear.

Answer

To show that the points A(1, 2, 1), B(2, 1, 0), and C(7/5, 8/5, 3/5) are collinear using the section formula, we will follow these steps:

### Step 1: Understand the Section Formula
The section formula states that if a point C divides the line segment joining points A(x1, y1, z1) and B(x2, y2, z2) in the ratio λ:1, then the coordinates of point C can be expressed as:
\[ C\left(\frac{\lambda x_2 + x_1}{\lambda + 1}, \frac{\lambda y_2 + y_1}{\lambda + 1}, \frac{\lambda z_2 + z_1}{\lambda + 1}\right) \]

### Step 2: Assign Coordinates
Let:
- A(1, 2, 1) = (x1, y1, z1)
- B(2, 1, 0) = (x2, y2, z2)
- C(7/5, 8/5, 3/5) = (x, y, z)

### Step 3: Apply the Section Formula
Using the section formula, we can express the coordinates of point C in terms of λ:
\[ C\left(\frac{\lambda \cdot 2 + 1}{\lambda + 1}, \frac{\lambda \cdot 1 + 2}{\lambda + 1}, \frac{\lambda \cdot 0 + 1}{\lambda + 1}\right) \]
This simplifies to:
\[ C\left(\frac{2\lambda + 1}{\lambda + 1}, \frac{\lambda + 2}{\lambda + 1}, \frac{1}{\lambda + 1}\right) \]

### Step 4: Set Up Equations
Now we will set the coordinates of C equal to the coordinates derived from the section formula:
1. For the x-coordinate:
\[ \frac{2\lambda + 1}{\lambda + 1} = \frac{7}{5} \]
2. For the y-coordinate:
\[ \frac{\lambda + 2}{\lambda + 1} = \frac{8}{5} \]
3. For the z-coordinate:
\[ \frac{1}{\lambda + 1} = \frac{3}{5} \]

### Step 5: Solve for λ
**For x-coordinate:**
Cross-multiplying gives:
\[ 5(2\lambda + 1) = 7(\lambda + 1) \]
Expanding:
\[ 10\lambda + 5 = 7\lambda + 7 \]
Rearranging:
\[ 3\lambda = 2 \implies \lambda = \frac{2}{3} \]

**For y-coordinate:**
Cross-multiplying gives:
\[ 5(\lambda + 2) = 8(\lambda + 1) \]
Expanding:
\[ 5\lambda + 10 = 8\lambda + 8 \]
Rearranging:
\[ 3\lambda = 2 \implies \lambda = \frac{2}{3} \]

**For z-coordinate:**
Cross-multiplying gives:
\[ 5 = 3(\lambda + 1) \]
Expanding:
\[ 5 = 3\lambda + 3 \]
Rearranging:
\[ 3\lambda = 2 \implies \lambda = \frac{2}{3} \]

### Step 6: Conclusion
Since we obtained the same value of λ for all three coordinates, we conclude that points A, B, and C are collinear.