When a right handed rectangular Cartesian system OXYZ is rotated about the z-axis through an angle `pi/4` in the counter-clockwise, direction it is found that a vector `veca` has the component `2sqrt(3), 3sqrt(2)` and 4.

Options

A. (a) '5,-1,4'
B. (b) `5,-1, 4sqrt(2)`
C. (c) `-1, -5, 4sqrt(2)`
D. (d) None of these

Answer

To solve the problem step by step, we need to find the components of the vector \(\vec{A}\) after the rotation of the coordinate system about the z-axis by an angle of \(\frac{\pi}{4}\) (45 degrees).

### Step 1: Write the initial vector components
The vector \(\vec{A}\) is given with components:
\[ \vec{A} = 2\sqrt{2} \hat{i} + 3\sqrt{2} \hat{j} + 4 \hat{k} \]

### Step 2: Determine the new unit vectors after rotation
When the coordinate system is rotated about the z-axis, the new unit vectors \(\hat{i_1}\), \(\hat{j_1}\), and \(\hat{k_1}\) can be expressed in terms of the original unit vectors \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\):
- The new \(\hat{i_1}\) after rotation is:
\[ \hat{i_1} = \cos\left(\frac{\pi}{4}\right) \hat{i} + \sin\left(\frac{\pi}{4}\right) \hat{j} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} \]
- The new \(\hat{j_1}\) after rotation is:
\[ \hat{j_1} = -\sin\left(\frac{\pi}{4}\right) \hat{i} + \cos\left(\frac{\pi}{4}\right) \hat{j} = -\frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} \]
- The new \(\hat{k_1}\) remains the same as \(\hat{k}\):
\[ \hat{k_1} = \hat{k} \]

### Step 3: Substitute the new unit vectors into the vector \(\vec{A}\)
Now we substitute the new unit vectors into the expression for \(\vec{A}\):
\[ \vec{A} = 2\sqrt{2} \hat{i_1} + 3\sqrt{2} \hat{j_1} + 4 \hat{k_1} \]
Substituting the expressions for \(\hat{i_1}\), \(\hat{j_1}\), and \(\hat{k_1}\):
\[ \vec{A} = 2\sqrt{2} \left(\frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j}\right) + 3\sqrt{2} \left(-\frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j}\right) + 4 \hat{k} \]

### Step 4: Simplify the expression
Now we simplify the expression:
\[ \vec{A} = 2 \hat{i} + 2 \hat{j} - 3 \hat{i} + 3 \hat{j} + 4 \hat{k} \]
Combining like terms:
\[ \vec{A} = (2 - 3) \hat{i} + (2 + 3) \hat{j} + 4 \hat{k} \]
\[ \vec{A} = -1 \hat{i} + 5 \hat{j} + 4 \hat{k} \]

### Step 5: Write the final components
Thus, the components of the vector \(\vec{A}\) after the rotation are:
\[ \text{Components: } -1, 5, 4 \]

### Final Answer
The components of the vector \(\vec{A}\) in the new coordinate system are:
\[ (-1, 5, 4) \]