If `vec(a) = 2 hat(i) - 3hat(j) + 2 hat(k), vec(b)= 2hat(i)-4 hat(k)` and `vec(c ) = hat(i) + lambda hat(j) + 3 hat(k)` are coplanar , then the values of `lambda ` is

Options

A. `-(5)/(2)`
B. `(5)/(3)`
C. `(3)/(5)`
D. `- ( 5)/(3)`

Answer

To solve the problem, we need to find the value of \( \lambda \) such that the vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) are coplanar. The vectors are given as:

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

### Step 1: Set up the determinant for coplanarity
Vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) are coplanar if the determinant of the matrix formed by these vectors is zero. We can express this as:

\[ \begin{vmatrix} 2 & -3 & 2 \\ 2 & 0 & -4 \\ 1 & \lambda & 3 \end{vmatrix} = 0 \]

### Step 2: Calculate the determinant
We will calculate the determinant using the formula for a 3x3 matrix:

\[ \text{Det} = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix, we have:

\[ \text{Det} = 2 \begin{vmatrix} 0 & -4 \\ \lambda & 3 \end{vmatrix} - (-3) \begin{vmatrix} 2 & -4 \\ 1 & 3 \end{vmatrix} + 2 \begin{vmatrix} 2 & 0 \\ 1 & \lambda \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. \( \begin{vmatrix} 0 & -4 \\ \lambda & 3 \end{vmatrix} = (0)(3) - (-4)(\lambda) = 4\lambda \)
2. \( \begin{vmatrix} 2 & -4 \\ 1 & 3 \end{vmatrix} = (2)(3) - (-4)(1) = 6 + 4 = 10 \)
3. \( \begin{vmatrix} 2 & 0 \\ 1 & \lambda \end{vmatrix} = (2)(\lambda) - (0)(1) = 2\lambda \)

Putting it all together:

\[ \text{Det} = 2(4\lambda) + 3(10) + 2(2\lambda) \]
\[ = 8\lambda + 30 + 4\lambda \]
\[ = 12\lambda + 30 \]

### Step 3: Set the determinant to zero
For coplanarity, we set the determinant to zero:

\[ 12\lambda + 30 = 0 \]

### Step 4: Solve for \( \lambda \)
Now, we solve for \( \lambda \):

\[ 12\lambda = -30 \]
\[ \lambda = -\frac{30}{12} = -\frac{5}{2} \]

### Conclusion
The value of \( \lambda \) such that the vectors are coplanar is:

\[ \lambda = -\frac{5}{2} \]