Statement 1 : Let `A(veca), B(vecb) and C(vecc)` be three points such that `veca = 2hati +hatk , vecb = 3hati -hatj +3hatk and vecc =-hati +7hatj -5hatk`. Then OABC is tetrahedron.
Statement 2 : Let `A(veca) , B(vecb) and C(vecc)` be three points such that vectors `veca, vecb and vecc` are non-coplanar. Then OABC is a tetrahedron, where O is the origin.

Options

A. Both the statements are true, and Statement 2 is the correct explanation for Statement 1.
B. Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.
C. Statement 1 is true and Statement 2 is false.
D. Statement 1 is false and Statement 2 is true.

Explanation

To determine whether the statements about the tetrahedron OABC are true, we need to analyze the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) given in the problem.

### Step 1: Identify the vectors
We have the following vectors:
- \(\vec{a} = 2\hat{i} + \hat{k}\)
- \(\vec{b} = 3\hat{i} - \hat{j} + 3\hat{k}\)
- \(\vec{c} = -\hat{i} + 7\hat{j} - 5\hat{k}\)

### Step 2: Form the matrix for the determinant
To check if the vectors are coplanar, we can form a matrix using the coefficients of the vectors and calculate the determinant. The vectors can be represented in a matrix as follows:

\[ \begin{vmatrix} 2 & 0 & 1 \\ 3 & -1 & 3 \\ -1 & 7 & -5 \end{vmatrix} \]

### Step 3: Calculate the determinant
We will calculate the determinant of the above matrix:

\[ D = 2 \begin{vmatrix} -1 & 3 \\ 7 & -5 \end{vmatrix} - 0 + 1 \begin{vmatrix} 3 & -1 \\ -1 & 7 \end{vmatrix} \]

Calculating the first determinant:

\[ \begin{vmatrix} -1 & 3 \\ 7 & -5 \end{vmatrix} = (-1)(-5) - (3)(7) = 5 - 21 = -16 \]

Calculating the second determinant:

\[ \begin{vmatrix} 3 & -1 \\ -1 & 7 \end{vmatrix} = (3)(7) - (-1)(-1) = 21 - 1 = 20 \]

Now substituting back into the determinant \(D\):

\[ D = 2(-16) + 0 + 1(20) = -32 + 20 = -12 \]

### Step 4: Conclusion about coplanarity
Since the determinant \(D = -12\) is not equal to zero, the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are non-coplanar.

### Step 5: Confirm the statements
- **Statement 1**: OABC is a tetrahedron because the vectors are non-coplanar.
- **Statement 2**: This statement is also true as it states that if the vectors are non-coplanar, OABC is a tetrahedron.

Thus, both statements are true.