Q.Consider the quantities, pressure, power, energy, impulse, gravitational potential, electrical charge, temperature, area. Out of these, the only vector quantities are

The key idea is to distinguish between scalar quantities, which are fully described by magnitude alone, and vector quantities, which require both magnitude and direction for their complete description.

1. Pressure, power, energy, gravitational potential, electrical charge, and temperature are all scalar quantities, as they are defined solely by their magnitude.
2. Impulse is defined as the change in momentum ($\Delta \vec{p} = \vec{F} \Delta t$). Since momentum ($\vec{p} = m\vec{v}$) is a vector quantity, impulse is also a vector quantity, having both magnitude and direction. …

In physics, quantities are broadly classified into two types: scalars and vectors. Understanding this distinction is fundamental because it dictates how these quantities are combined and manipulated mathematically.

A scalar quantity is completely defined by its magnitude alone. It does not have a direction associated with it. Examples include mass, time, distance, speed, and temperature. Scalar quantities are added, subtracted, multiplied, and divided using ordinary arithmetic rules.

A vector quantity, on the other hand, is defined by both its magnitude and its direction. Additionally, vector quantities must obey the rules of vector addition (e.g., triangle law or parallelogram law of vector addition). Examples include displacement, velocity, acceleration, and force.

Let's analyze each quantity provided in the list:

1. Pressure: Pressure is defined as force per unit area ($P = F/A$). While force is a vector, pressure itself is a scalar quantity. At any point within a fluid, pressure acts equally in all directions. It does not have a specific direction in space. For instance, the pressure inside a balloon is a single value, not a directional quantity.

2. Power: Power is the rate at which work is done or energy is transferred ($P = W/t$). Both work and energy are scalar quantities, and time is also a scalar. Therefore, power is a scalar quantity. It only has a magnitude, indicating how quickly energy is being used or produced.

3. Energy: Energy is the capacity to do work. It exists in various forms (kinetic, potential, thermal, etc.) but is always a scalar quantity. For example, a body has a certain amount of kinetic energy, but this energy does not point in a particular direction.

4. Impulse: Impulse ($\vec{J}$) is defined as the change in momentum ($\Delta \vec{p}$) of an object. Momentum ($\vec{p} = m\vec{v}$) is a vector quantity because it depends on velocity, which is a vector. Since impulse is a change in a vector quantity, impulse itself is a vector quantity. Its direction is the same as the direction of the net force causing the change in momentum.

   $\vec{J} = \Delta \vec{p} = \vec{F}_{avg} \Delta t$

5. Gravitational Potential: Gravitational potential ($V_g$) at a point is defined as the work done per unit mass to bring a test mass from infinity to that point. Since work is a scalar quantity and mass is a scalar quantity, gravitational potential is a scalar quantity. It represents a scalar field in space.

6. Electrical Charge: Electrical charge ($q$) is a fundamental property of matter. It can be positive or negative, but it does not have a direction in space. Therefore, electrical charge is a scalar quantity.

7. Temperature: Temperature ($T$) is a measure of the average kinetic energy of the particles within a substance. It indicates the degree of hotness or coldness. Temperature is a scalar quantity; it only has a magnitude and no direction. …

Concept: Trace Every Quantity to Its Defining Formula, Not Its Physical Picture

Method: Formula-Tracing Decision Table (Dot Product $\Rightarrow$ Scalar, Cross/Scalar-Multiple $\Rightarrow$ Vector)

Rather than reasoning in prose about whether each of the eight named quantities "has a direction," this method writes down the defining relation of each one and applies a single mechanical rule: a quantity built as a dot product of two vectors (or as a pure ratio/count of scalars) collapses to a scalar; a quantity built as a cross product, or as a scalar times a vector, survives as a vector.

The rule

• Scalar $\times$ scalar $\to$ scalar. Scalar $\times$ vector $\to$ vector (rescales it, keeps its direction).
• $\vec u \cdot \vec v$ (dot product) $\to$ always a scalar, regardless of $\vec u,\vec v$.
• $\vec u \times \vec v$ (cross product), or an oriented area/surface, $\to$ a vector, perpendicular to the plane spanned.

Applying the rule to each quantity