To determine the number of distinct real values of \(\lambda\) for which the vectors \(\vec{a} = \lambda^3 \hat{i} + \hat{k}\), \(\vec{b} = \hat{i} - \lambda^3 \hat{j}\), and \(\vec{c} = \hat{i} + (2\lambda - \sin \lambda) \hat{j} - \lambda \hat{k}\) are coplanar, we can use the condition that the scalar triple product of the vectors must be zero.
### Step-by-Step Solution:
1. **Write the vectors in component form**:
\[
\vec{a} = \begin{pmatrix} \lambda^3 \\ 0 \\ 1 \end{pmatrix}, \quad \vec{b} = \begin{pmatrix} 1 \\ -\lambda^3 \\ 0 \end{pmatrix}, \quad \vec{c} = \begin{pmatrix} 1 \\ 2\lambda - \sin \lambda \\ -\lambda \end{pmatrix}
\]
2. **Set up the matrix for the scalar triple product**:
The scalar triple product can be represented as the determinant of the matrix formed by the vectors:
\[
\text{Det} \begin{pmatrix}
\lambda^3 & 0 & 1 \\
1 & -\lambda^3 & 0 \\
1 & 2\lambda - \sin \lambda & -\lambda
\end{pmatrix}
\]
3. **Calculate the determinant**:
Using the determinant formula for a 3x3 matrix:
\[
\text{Det} = a(ei - fh) - b(di - fg) + c(dh - eg)
\]
where \(a, b, c\) are the first row elements, \(d, e, f\) are the second row elements, and \(g, h, i\) are the third row elements.
Here:
- \(a = \lambda^3\), \(b = 0\), \(c = 1\)
- \(d = 1\), \(e = -\lambda^3\), \(f = 0\)
- \(g = 1\), \(h = 2\lambda - \sin \lambda\), \(i = -\lambda\)
The determinant becomes:
\[
\text{Det} = \lambda^3((- \lambda)(2\lambda - \sin \lambda) - 0) - 0 + 1(1 \cdot 0 - (-\lambda^3)(2\lambda - \sin \lambda))
\]
Simplifying:
\[
= -\lambda^4 + \lambda^3(2\lambda - \sin \lambda)
\]
\[
= \lambda^3(2\lambda - \sin \lambda - \lambda)
\]
\[
= \lambda^3(\lambda - \sin \lambda)
\]
4. **Set the determinant to zero for coplanarity**:
\[
\lambda^3(\lambda - \sin \lambda) = 0
\]
This gives us two cases:
- \(\lambda^3 = 0 \Rightarrow \lambda = 0\)
- \(\lambda - \sin \lambda = 0\)
5. **Solve \(\lambda - \sin \lambda = 0\)**:
The equation \(\lambda = \sin \lambda\) has solutions where \(\lambda\) intersects the sine curve. The function \(\sin \lambda\) oscillates between -1 and 1, while \(\lambda\) is a straight line. The intersections occur at:
- \(\lambda = 0\)
- Other intersections can be found graphically or numerically.
The function \(\lambda - \sin \lambda\) has a slope greater than 1 for \(\lambda > 0\) and less than -1 for \(\lambda < 0\), indicating that there will be exactly one intersection in each interval where \(\sin\) oscillates.
6. **Count the distinct solutions**:
- The solution \(\lambda = 0\) counts as one distinct solution.
- The equation \(\lambda = \sin \lambda\) has two other distinct solutions in the intervals \((0, \pi/2)\) and \((-\pi/2, 0)\).
Thus, the total number of distinct real values of \(\lambda\) for which the vectors are coplanar is **3**.
