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Q2(iv):
Classify the following measures as scalars and vectors.
(iv) 40 watt
Solution :
1. Definitions & Conceptual Framework
To classify a physical quantity, we must distinguish between scalars and vectors based on their mathematical properties:
- Scalar: A physical quantity characterized solely by its magnitude. It requires a numerical value and a unit of measurement. It is independent of direction [Per Scalar Field Theory].
- Vector: A physical quantity characterized by both magnitude and a specific direction in space. It must satisfy the laws of vector addition [Per Euclidean Vector Space definitions].
2. Analysis of the Physical Quantity
The quantity provided is $40\text{ watt}$.
- Magnitude: $40$
- Unit: $\text{watt (W)}$, which is the SI unit of power.
- Dimensional Analysis: Power $P$ is defined as the rate of energy transfer or work done per unit time: $P = \frac{dW}{dt}$ [Per the Definition of Power in Classical Mechanics].
3. Logical Deduction
Power is calculated as the scalar product of force $\vec{F}$ and velocity $\vec{v}$ ($P = \vec{F} \cdot \vec{v}$). Although force and velocity are vectors, the result of a dot product is always a scalar quantity [By the Properties of the Scalar Product]. Since $40\text{ watt}$ describes the rate of energy conversion without reference to a directional component, it satisfies the criteria for a scalar quantity.
4. Visual Representation of Scalar Power
Final Solution: The measure 40 watt 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(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
- Q3(v): Classify the following as scalar and vector quantities. (v) work done
- 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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