A vector A has components `A_(1), A_(2), A_(3)` along the co-ordinate axes respectively. The co-ordinate system is rotated about Z-axis through an angle `pi//2`with anticlockwise direction. Then the components of A in new co-ordinate system are ...............

Answer

To solve the problem of finding the new components of vector **A** after rotating the coordinate system about the Z-axis through an angle of \(\frac{\pi}{2}\) (90 degrees) in an anticlockwise direction, we can follow these steps:

### Step 1: Understand the Initial Components
The vector **A** has components:
- \(A_1\) along the X-axis
- \(A_2\) along the Y-axis
- \(A_3\) along the Z-axis

### Step 2: Visualize the Rotation
When we rotate the coordinate system about the Z-axis:
- The Z-axis remains unchanged.
- The X-axis and Y-axis will switch positions.

### Step 3: Determine the New X-axis Component
After a \(\frac{\pi}{2}\) rotation:
- The original X-axis (which had the component \(A_1\)) will now align with the Y-axis.
- Therefore, the new X-axis component will be equal to the original Y-axis component, which is \(A_2\).

### Step 4: Determine the New Y-axis Component
- The original Y-axis (which had the component \(A_2\)) will now align with the negative X-axis.
- Thus, the new Y-axis component will be the negative of the original X-axis component, which is \(-A_1\).

### Step 5: Determine the New Z-axis Component
- The Z-axis remains unchanged, so the Z-axis component will still be \(A_3\).

### Step 6: Write the New Components
Putting it all together, the components of vector **A** in the new coordinate system after the rotation are:
- New X-component: \(A_2\)
- New Y-component: \(-A_1\)
- New Z-component: \(A_3\)

Thus, the new components of vector **A** are:
\[ \text{New components of } \mathbf{A} = (A_2, -A_1, A_3) \]

### Final Answer
The components of vector **A** in the new coordinate system are:
\[ (A_2, -A_1, A_3) \]

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