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Q3(v):
Classify the following as scalar and vector quantities.
(v) work done
Solution :
Theoretical Definition of the Physical Quantity
In classical mechanics, work done ($W$) is defined as the product of the magnitude of the force applied to an object and the magnitude of the displacement in the direction of that force. Mathematically, it is expressed as the scalar product (or dot product) of the force vector ($\vec{F}$) and the displacement vector ($\vec{d}$):
$W = \vec{F} \cdot \vec{d} = |\vec{F}| |\vec{d}| \cos(\theta)$
[Per the definition of the Dot Product in Vector Algebra: The dot product of two vectors always yields a scalar value, as $\cos(\theta)$ is a real number ranging from $-1$ to $1$.]
Step 1: Analyzing the Nature of the Quantity
- Magnitude: Work done possesses a magnitude (e.g., $50 \text{ Joules}$), which accounts for the energy transferred.
- Direction: Work does not have a direction in space. While a force vector has a direction and a displacement vector has a direction, their product—work—represents the scalar outcome of their interaction.
- Algebraic Behavior: Work can be positive, negative, or zero based on the angle $\theta$ between the force and displacement vectors. However, being negative does not imply directionality in the vector sense; it merely indicates energy transfer direction relative to the system.
Step 2: Graphical Representation of the Dot Product
Step 3: Classification Justification
Since work done $W$ is defined by the dot product of two vectors, it lacks a direction. In vector calculus, quantities defined by the projection of one vector onto another, resulting in a magnitude only, are strictly classified as Scalars.
| Criterion | Result |
|---|---|
| Possesses Magnitude? | Yes |
| Possesses Direction? | No |
| Obeys Vector Addition? | N/A (Scalar Algebra applies) |
Final Solution: Work done is a scalar quantity.
More Questions from Class 12 Mathematics Vector Algebra EXERCISE 10.1
- Q1: Represent graphically a displacement of 40 km, 30° east of north.
- Q2(i): Classify the following measures as scalars and vectors. (i) 10 kg
- Q2(ii): Classify the following measures as scalars and vectors. (ii) 2 meters north-west
- Q2(iii): Classify the following measures as scalars and vectors. (iii) 40°
- Q2(iv): Classify the following measures as scalars and vectors. (iv) 40 watt
- Q2(v): Classify the following measures as scalars and vectors. (v) $10^{-19}$ coulomb
- Q2(vi): Classify the following measures as scalars and vectors. (vi) $20$ m/s$^2$
- Q3(i): Classify the following as scalar and vector quantities. (i) time period
- Q3(ii): Classify the following as scalar and vector quantities. (ii) distance
- Q3(iii): Classify the following as scalar and vector quantities. (iii) force
- Q3(iv): Classify the following as scalar and vector quantities. (iv) velocity
- Q4(i): In Fig 10.6 (a square), identify the following vectors. (i) Coinitial
- Q4(ii): In Fig 10.6 (a square), identify the following vectors. (ii) Equal
- Q4(iii): In Fig 10.6 (a square), identify the following vectors. (iii) Collinear but not equal
- Q5(i): Answer the following as true or false. (i) $\vec{a}$ and $-\vec{a}$ are collinear.
- Q5(ii): Answer the following as true or false. (ii) Two collinear vectors are always equal in magnitude.
- Q5(iii): Answer the following as true or false. (iii) Two vectors having same magnitude are collinear.
- Q5(iv): Answer the following as true or false. (iv) Two collinear vectors having the same magnitude are equal.
CBSE Solutions for Class 12 Mathematics Vector Algebra
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