Find a vector of magnitude 7 units, which is perpendicular to two vectors :
`2hat(i)-hat(j)+hat(k)` and `hat(i)+hat(j)-hat(k)`.

Answer

To find a vector of magnitude 7 units that is perpendicular to the vectors \( \mathbf{A} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \mathbf{B} = \hat{i} + \hat{j} - \hat{k} \), we can follow these steps:

### Step 1: Find the Cross Product of Vectors A and B
The vector perpendicular to both \( \mathbf{A} \) and \( \mathbf{B} \) can be found using the cross product \( \mathbf{C} = \mathbf{A} \times \mathbf{B} \).

\[ \mathbf{C} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -1 & 1 \\ 1 & 1 & -1 \end{vmatrix} \]

### Step 2: Calculate the Determinant
Calculating the determinant, we expand it as follows:

\[ \mathbf{C} = \hat{i} \begin{vmatrix} -1 & 1 \\ 1 & -1 \end{vmatrix} - \hat{j} \begin{vmatrix} 2 & 1 \\ 1 & -1 \end{vmatrix} + \hat{k} \begin{vmatrix} 2 & -1 \\ 1 & 1 \end{vmatrix} \]

Calculating each of these 2x2 determinants:

1. For \( \hat{i} \):
\[ (-1)(-1) - (1)(1) = 1 - 1 = 0 \]

2. For \( \hat{j} \):
\[ (2)(-1) - (1)(1) = -2 - 1 = -3 \]

3. For \( \hat{k} \):
\[ (2)(1) - (-1)(1) = 2 + 1 = 3 \]

Putting it all together, we have:
\[ \mathbf{C} = 0\hat{i} + 3\hat{j} + 3\hat{k} = 3\hat{j} + 3\hat{k} \]

### Step 3: Find the Magnitude of Vector C
Now, we need to find the magnitude of vector \( \mathbf{C} \):
\[ |\mathbf{C}| = \sqrt{(0)^2 + (3)^2 + (3)^2} = \sqrt{0 + 9 + 9} = \sqrt{18} = 3\sqrt{2} \]

### Step 4: Find the Unit Vector in the Direction of C
To find the unit vector \( \hat{C} \):
\[ \hat{C} = \frac{\mathbf{C}}{|\mathbf{C}|} = \frac{3\hat{j} + 3\hat{k}}{3\sqrt{2}} = \frac{\hat{j} + \hat{k}}{\sqrt{2}} \]

### Step 5: Find the Required Vector D of Magnitude 7
To find the vector \( \mathbf{D} \) of magnitude 7 in the direction of \( \hat{C} \):
\[ \mathbf{D} = 7 \hat{C} = 7 \cdot \frac{\hat{j} + \hat{k}}{\sqrt{2}} = \frac{7}{\sqrt{2}} \hat{j} + \frac{7}{\sqrt{2}} \hat{k} \]

### Final Answer
Thus, the vector of magnitude 7 units that is perpendicular to both given vectors is:
\[ \mathbf{D} = \frac{7}{\sqrt{2}} \hat{j} + \frac{7}{\sqrt{2}} \hat{k} \]