Let A ? be a unit vector along the axis of rotation of a purely rotating body and B ? be a unit vector along the velocity of a particle P of the body away from the axis. The value of A ? .B ? is

To determine the value of the dot product \( A \cdot B \) where \( A \) is a unit vector along the axis of rotation of a purely rotating body and \( B \) is a unit vector along the velocity of a particle \( P \) moving away from that axis, we need to consider the geometric relationship between these two vectors.

Understanding the Vectors

First, let's clarify what each vector represents:

• Vector A: This is a unit vector that points along the axis of rotation. Since it is a unit vector, its magnitude is 1.
• Vector B: This is also a unit vector, but it points in the direction of the velocity of particle \( P \). As the particle moves away from the axis of rotation, its velocity vector will be perpendicular to the axis of rotation in a purely rotating system.

Geometric Interpretation

In a purely rotating body, the velocity of any point on the body is always perpendicular to the radius vector that extends from the axis of rotation to that point. Therefore, the velocity vector \( B \) is orthogonal to the axis of rotation represented by vector \( A \).

Calculating the Dot Product

The dot product of two vectors \( A \) and \( B \) is defined mathematically as:

A · B = |A| |B| cos(θ)

Where \( θ \) is the angle between the two vectors. Since both \( A \) and \( B \) are unit vectors, their magnitudes \( |A| \) and \( |B| \) are both equal to 1. Thus, the equation simplifies to:

A · B = cos(θ)

Determining the Angle

In our scenario, since \( A \) (the axis of rotation) and \( B \) (the velocity vector) are perpendicular to each other, the angle \( θ \) between them is 90 degrees. The cosine of 90 degrees is 0:

cos(90°) = 0

Final Result

Putting it all together, we find:

A · B = 0

Thus, the value of the dot product \( A \cdot B \) is 0, indicating that the two vectors are orthogonal. This result is consistent with the physical understanding of rotational motion, where the velocity of a point on a rotating body is always perpendicular to the axis of rotation.