Q.Classify the following measures as scalars and vectors.

Rule: a scalar has magnitude only; a vector has magnitude and a direction.

1. $10$ kg — mass $\rightarrow$ scalar
2. $2$ metres north-west — has a direction $\rightarrow$ vector
3. $40^\circ$ — a bare angle $\rightarrow$ scalar
4. $40$ watt — power $\rightarrow$ scalar
5. $10^{-19}$ coulomb — electric charge $\rightarrow$ scalar
6. $20$ m/s$^2$ — acceleration, a directed quantity $\rightarrow$ vector
   ✓Final answer

   Scalars: (i), (iii), (iv), (v). Vectors: (ii) and (vi).

The test

For each measure ask: does a direction belong to it? If yes, it is a vector; if the number-with-unit is complete on its own, it is a scalar.

1. $10$ kg — mass, purely a magnitude. Scalar.
2. $2$ metres north-west — a length together with a stated direction (north-west); a displacement. Vector.
3. $40^\circ$ — an angle; just a number of degrees with no direction of its own. Scalar.
4. $40$ watt — power, the rate of doing work; magnitude only. Scalar.
5. $10^{-19}$ coulomb — electric charge; a scalar (its $+$/$-$ sign is polarity, not a spatial direction). Scalar.
6. $20$ m/s$^2$ — acceleration, which is defined as a vector: it always acts in a definite direction (that of the net force). Vector.

Method: Classifying a quantity as scalar or vector

Use this for any "classify as scalar/vector" question — decide by whether a genuine spatial direction belongs to the quantity.

Steps

Step 1: Ask the one diagnostic question

Does the quantity need a direction in space to be fully specified, or is a number-with-unit complete on its own?

• Direction required $\Rightarrow$ vector.
• Magnitude alone suffices $\Rightarrow$ scalar.

Step 2: Separate a sign from a direction

A $+$/$-$ sign (charge polarity, a temperature below zero) is not a spatial direction. Such quantities stay scalars.

Step 3: Watch the physics of each item

Displacement, velocity, acceleration and force are directed (vectors); mass, time, power, charge, energy/work, angle and distance are not (scalars). Apply the test item by item rather than guessing from the unit alone.

Common Mistakes

Mistake 1: Calling electric charge a vector because it can be negative

Why it's wrong: the sign of a charge is polarity, not a direction in space. Correct approach: charge is a scalar; only a genuine spatial direction (like "north-west") makes a quantity a vector.

Mistake 2: Treating an angle like a direction

Why it's wrong: "$40^\circ$" is just a magnitude of rotation with no direction of its own, so it is a scalar. Correct approach: classify a bare angle as scalar.

Mistake 3: Overlooking that acceleration is a vector

Why it's wrong: "$20\ \text{m/s}^2$" always acts in the direction of the net force, so it is a vector even when written as a bare number. Correct approach: recall acceleration, like velocity and force, is directed.