Rotational kinematics — which of the following are vector quantities? Select the correct set when considering direction-sensitive rotational measures.

Introduction / Context:
In rotational motion, many quantities possess both magnitude and direction (sense about an axis), making them vectors (more precisely, axial vectors or pseudovectors). Distinguishing vector from scalar quantities is essential for applying vector algebra and sign conventions correctly.

Given Data / Assumptions:

• Standard rigid-body rotation about a fixed axis through small angles.
• Right-hand rule used to assign vector directions.

Concept / Approach:
Angular displacement, angular velocity, and angular acceleration each require a direction specification (clockwise vs counter-clockwise mapped to an axis via the right-hand rule). Hence they are treated as vectors in engineering analyses. Their directions are along the axis of rotation: thumb points along +ω when fingers curl in the rotation sense.

Step-by-Step Solution:
Angular displacement: for small rotations, represented as a vector along the axis (direction matters in composition).Angular velocity ω: vector indicating rotation rate and axis direction; used in v = ω × r.Angular acceleration α: time rate of change of ω; direction indicates speeding up or slowing down and precessional effects.Therefore, all listed quantities are vectors.

Verification / Alternative check:
Use vector relationships: linear velocity v at a point is v = ω × r, which only makes sense if ω is a vector. Similarly, tangential acceleration a_t = α × r requires vector α.

Why Other Options Are Wrong:

• Selecting only one or two ignores the consistent vector treatment used in kinematics and dynamics.

Common Pitfalls:

• Believing angular displacement is always scalar; for finite large rotations, non-commutativity arises, but in engineering practice small-angle vector treatment is standard.

Final Answer:
all of these