Statement-1 Unit vector Orthogonal to `5hati+2hatj+6hatk` are coplanar with `2hati+hatj+hatk` and `hati-hatj+hatk` is `(3hatj-hatk)/sqrt10`.
Statement -2 IF `bara,barb,barc` represents the vector along coterminus edge of parallelepiped then `|bara.(barb times barc)|` represents the volume

Options

A. Statement -1 is true, Statement -2 is true, Statement-2 is a correct explanation for Statement -1
B. Statement-1 is true,Statement-2 is true,Statement-2is NOT a correct explanation for Statement-1
C. Statement-1 is True,Statement-2 is false
D. Statement-1 is false, Statement-2 is true

Answer

To solve the problem step by step, we will analyze both statements provided in the question.

### Step 1: Analyze Statement 1
We need to find a unit vector orthogonal to the vector \( \mathbf{A} = 5\hat{i} + 2\hat{j} + 6\hat{k} \) that is also coplanar with the vectors \( \mathbf{B} = 2\hat{i} + \hat{j} + \hat{k} \) and \( \mathbf{C} = \hat{i} - \hat{j} + \hat{k} \).

#### Step 1.1: Find the orthogonal vector
A vector \( \mathbf{D} \) is orthogonal to \( \mathbf{A} \) if:
\[ \mathbf{A} \cdot \mathbf{D} = 0 \]
Assuming \( \mathbf{D} = x\hat{i} + y\hat{j} + z\hat{k} \), we have:
\[ 5x + 2y + 6z = 0 \quad \text{(1)} \]

#### Step 1.2: Coplanarity condition
The vectors \( \mathbf{B} \), \( \mathbf{C} \), and \( \mathbf{D} \) are coplanar if:
\[ \mathbf{D} = \lambda \mathbf{B} + \mu \mathbf{C} \]
Substituting the vectors:
\[ \mathbf{D} = \lambda(2\hat{i} + \hat{j} + \hat{k}) + \mu(\hat{i} - \hat{j} + \hat{k}) \]
This gives:
\[ \mathbf{D} = (2\lambda + \mu)\hat{i} + (\lambda - \mu)\hat{j} + (\lambda + \mu)\hat{k} \]

#### Step 1.3: Equate components
From the expression for \( \mathbf{D} \), we can equate components with \( x, y, z \):
- \( x = 2\lambda + \mu \) (2)
- \( y = \lambda - \mu \) (3)
- \( z = \lambda + \mu \) (4)

#### Step 1.4: Substitute into equation (1)
Substituting equations (2), (3), and (4) into equation (1):
\[ 5(2\lambda + \mu) + 2(\lambda - \mu) + 6(\lambda + \mu) = 0 \]
Expanding this:
\[ 10\lambda + 5\mu + 2\lambda - 2\mu + 6\lambda + 6\mu = 0 \]
Combining like terms:
\[ (10 + 2 + 6)\lambda + (5 - 2 + 6)\mu = 0 \]
\[ 18\lambda + 9\mu = 0 \]
This implies:
\[ 2\lambda + \mu = 0 \quad \Rightarrow \quad \mu = -2\lambda \]

#### Step 1.5: Substitute back to find \( \mathbf{D} \)
Substituting \( \mu = -2\lambda \) back into equations (2), (3), and (4):
- From (2): \( x = 2\lambda - 2\lambda = 0 \)
- From (3): \( y = \lambda - (-2\lambda) = 3\lambda \)
- From (4): \( z = \lambda - 2\lambda = -\lambda \)

Thus, \( \mathbf{D} = 0\hat{i} + 3\lambda\hat{j} - \lambda\hat{k} \).

#### Step 1.6: Normalize to find the unit vector
The unit vector \( \mathbf{D} \) can be expressed as:
\[ \mathbf{D} = 3\hat{j} - \hat{k} \]
The magnitude is:
\[ |\mathbf{D}| = \sqrt{0^2 + 3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \]
Thus, the unit vector is:
\[ \hat{D} = \frac{3\hat{j} - \hat{k}}{\sqrt{10}} \]

### Conclusion for Statement 1
Statement 1 is true: The unit vector orthogonal to \( 5\hat{i} + 2\hat{j} + 6\hat{k} \) and coplanar with the other two vectors is indeed \( \frac{3\hat{j} - \hat{k}}{\sqrt{10}} \).

### Step 2: Analyze Statement 2
Statement 2 states that if \( \bar{a}, \bar{b}, \bar{c} \) represent vectors along the edges of a parallelepiped, then \( |\bar{a} \cdot (\bar{b} \times \bar{c})| \) represents the volume.

#### Step 2.1: Understanding the volume of a parallelepiped
The volume \( V \) of a parallelepiped formed by vectors \( \bar{a}, \bar{b}, \bar{c} \) is given by the scalar triple product:
\[ V = |\bar{a} \cdot (\bar{b} \times \bar{c})| \]
This is a well-known result in vector calculus.

### Conclusion for Statement 2
Statement 2 is also true: The expression \( |\bar{a} \cdot (\bar{b} \times \bar{c})| \) indeed represents the volume of the parallelepiped formed by the vectors.

### Final Conclusion
Both statements are true.