For what value of `'lambda'` are the following vectors coplanar ?
`vec(a)=2hat(i)-4hat(j)+5hat(k), vec(b)=hat(i)-lambda hat(j)+hat(k)` and `vec(c )= 3hat(i)+2hat(j)-5hat(k)`.

Answer

To determine the value of \( \lambda \) for which the vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) are coplanar, we need to calculate the scalar triple product of these vectors and set it equal to zero. The vectors are given as:

\[ \vec{a} = 2\hat{i} - 4\hat{j} + 5\hat{k} \]
\[ \vec{b} = \hat{i} - \lambda \hat{j} + \hat{k} \]
\[ \vec{c} = 3\hat{i} + 2\hat{j} - 5\hat{k} \]

### Step 1: Set up the determinant

The scalar triple product can be expressed as the determinant of a matrix formed by the coefficients of the vectors:

\[ \begin{vmatrix} 2 & -4 & 5 \\ 1 & -\lambda & 1 \\ 3 & 2 & -5 \end{vmatrix} \]

### Step 2: Calculate the determinant

We will expand this determinant using the first row:

\[ = 2 \begin{vmatrix} -\lambda & 1 \\ 2 & -5 \end{vmatrix} - (-4) \begin{vmatrix} 1 & 1 \\ 3 & -5 \end{vmatrix} + 5 \begin{vmatrix} 1 & -\lambda \\ 3 & 2 \end{vmatrix} \]

### Step 3: Calculate each of the 2x2 determinants

1. For the first determinant:

\[ \begin{vmatrix} -\lambda & 1 \\ 2 & -5 \end{vmatrix} = (-\lambda)(-5) - (1)(2) = 5\lambda - 2 \]

2. For the second determinant:

\[ \begin{vmatrix} 1 & 1 \\ 3 & -5 \end{vmatrix} = (1)(-5) - (1)(3) = -5 - 3 = -8 \]

3. For the third determinant:

\[ \begin{vmatrix} 1 & -\lambda \\ 3 & 2 \end{vmatrix} = (1)(2) - (-\lambda)(3) = 2 + 3\lambda \]

### Step 4: Substitute back into the determinant expression

Now substituting back into the determinant expression:

\[ = 2(5\lambda - 2) + 4(-8) + 5(2 + 3\lambda) \]
\[ = 10\lambda - 4 - 32 + 10 + 15\lambda \]
\[ = 25\lambda - 26 \]

### Step 5: Set the determinant equal to zero

For the vectors to be coplanar, we set the determinant equal to zero:

\[ 25\lambda - 26 = 0 \]

### Step 6: Solve for \( \lambda \)

Solving for \( \lambda \):

\[ 25\lambda = 26 \]
\[ \lambda = \frac{26}{25} \]

### Final Answer

Thus, the value of \( \lambda \) for which the vectors are coplanar is:

\[ \lambda = \frac{26}{25} \]