A vector `a` has components ` a_1,a_2,a_3` in a right handed rectangular cartesian coordinate system `OXYZ` the coordinate axis is rotated about `z` axis through an angle `pi/2`. The components of `a` in the new system

Explanation

To solve the problem of finding the components of vector \( \mathbf{a} \) after a rotation about the \( z \)-axis through an angle of \( \frac{\pi}{2} \), we will consider two cases: anticlockwise rotation and clockwise rotation.

### Step-by-Step Solution:

#### Case 1: Anticlockwise Rotation

1. **Identify the Original Components**:
The original components of the vector \( \mathbf{a} \) are given as \( a_1, a_2, a_3 \).

2. **Understand the Rotation**:
When the coordinate system is rotated anticlockwise about the \( z \)-axis, the old \( x \)-axis moves to the position of the old \( y \)-axis, and the old \( y \)-axis moves to the position of the negative old \( x \)-axis.

3. **Determine New Components**:
After the rotation, the new components of the vector \( \mathbf{a} \) in the rotated coordinate system will be:
- The new \( x \)-component will be the old \( y \)-component: \( a_2 \)
- The new \( y \)-component will be the negative of the old \( x \)-component: \( -a_1 \)
- The \( z \)-component remains unchanged: \( a_3 \)

Therefore, the components in the new system after anticlockwise rotation are:
\[ \mathbf{a}_{\text{new}} = (a_2, -a_1, a_3) \]

#### Case 2: Clockwise Rotation

1. **Identify the Original Components**:
The original components of the vector \( \mathbf{a} \) are still \( a_1, a_2, a_3 \).

2. **Understand the Rotation**:
When the coordinate system is rotated clockwise about the \( z \)-axis, the old \( y \)-axis moves to the position of the old \( x \)-axis, and the old \( x \)-axis moves to the position of the negative old \( y \)-axis.

3. **Determine New Components**:
After the clockwise rotation, the new components of the vector \( \mathbf{a} \) in the rotated coordinate system will be:
- The new \( x \)-component will be the negative of the old \( y \)-component: \( -a_2 \)
- The new \( y \)-component will be the old \( x \)-component: \( a_1 \)
- The \( z \)-component remains unchanged: \( a_3 \)

Therefore, the components in the new system after clockwise rotation are:
\[ \mathbf{a}_{\text{new}} = (-a_2, a_1, a_3) \]

### Summary of Results

- For **anticlockwise rotation**: \( \mathbf{a}_{\text{new}} = (a_2, -a_1, a_3) \)
- For **clockwise rotation**: \( \mathbf{a}_{\text{new}} = (-a_2, a_1, a_3) \)