The points `(3,6,9), (1,2,3), (2,3,4)` and `(4,6,lambda)` are coplanar if `lambda`=_______.

Answer

To determine the value of \( \lambda \) for which the points \( (3,6,9), (1,2,3), (2,3,4) \), and \( (4,6,\lambda) \) are coplanar, we can use the condition that the scalar triple product of the vectors formed by these points must be zero.

### Step-by-step Solution:

1. **Define the Points:**
Let:
- Point A = \( (3, 6, 9) \)
- Point B = \( (1, 2, 3) \)
- Point C = \( (2, 3, 4) \)
- Point D = \( (4, 6, \lambda) \)

2. **Calculate the Vectors:**
We need to calculate the vectors \( \vec{AB} \), \( \vec{AC} \), and \( \vec{AD} \):
- \( \vec{AB} = B - A = (1 - 3, 2 - 6, 3 - 9) = (-2, -4, -6) \)
- \( \vec{AC} = C - A = (2 - 3, 3 - 6, 4 - 9) = (-1, -3, -5) \)
- \( \vec{AD} = D - A = (4 - 3, 6 - 6, \lambda - 9) = (1, 0, \lambda - 9) \)

3. **Set Up the Scalar Triple Product:**
The scalar triple product \( \vec{AB} \cdot (\vec{AC} \times \vec{AD}) \) must equal zero for the points to be coplanar. We can express this as a determinant:
\[ \begin{vmatrix} -2 & -4 & -6 \\ -1 & -3 & -5 \\ 1 & 0 & \lambda - 9 \end{vmatrix} = 0 \]

4. **Calculate the Determinant:**
We will calculate the determinant using the method of cofactor expansion:
\[ = -2 \begin{vmatrix} -3 & -5 \\ 0 & \lambda - 9 \end{vmatrix} - (-4) \begin{vmatrix} -1 & -5 \\ 1 & \lambda - 9 \end{vmatrix} - 6 \begin{vmatrix} -1 & -3 \\ 1 & 0 \end{vmatrix} \]

Now, calculating each of the 2x2 determinants:
- For the first determinant:
\[ = -2((-3)(\lambda - 9) - (0)(-5)) = -2(-3\lambda + 27) = 6\lambda - 54 \]
- For the second determinant:
\[ = -4((-1)(\lambda - 9) - (-5)(1)) = -4(-\lambda + 9 + 5) = -4(-\lambda + 14) = 4\lambda - 56 \]
- For the third determinant:
\[ = -6((-1)(0) - (-3)(1)) = -6(3) = -18 \]

5. **Combine the Results:**
Now combine these results:
\[ 6\lambda - 54 + 4\lambda - 56 - 18 = 0 \]
Simplifying this gives:
\[ 10\lambda - 128 = 0 \]

6. **Solve for \( \lambda \):**
\[ 10\lambda = 128 \implies \lambda = \frac{128}{10} = 12.8 \]

### Final Answer:
Thus, the value of \( \lambda \) for which the points are coplanar is \( \lambda = 12.8 \).

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