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Q2(iii):
Classify the following measures as scalars and vectors.
(iii) 40°
Solution :
Step 1: Defining the Measurement
The given measure is $40^\circ$. This represents a magnitude of temperature or angular measurement depending on the physical context, but inherently, it signifies a scalar value.
Step 2: Theoretical Criteria for Classification
To distinguish between scalar and vector quantities, we apply the following definitions:
- Scalar Quantity: A physical quantity that is completely described by its magnitude (a numerical value along with units) and possesses no specific direction in space [Per the definition of Scalar fields in Euclidean space].
- Vector Quantity: A physical quantity that requires both magnitude and a defined direction to be completely specified. It must also satisfy the parallelogram law of vector addition [Per the axiomatic definition of Vectors in $\mathbb{R}^n$].
Step 3: Logical Analysis of the Measure
The measurement $40^\circ$ is a pure magnitude. Whether it represents an angle between two lines or a temperature on a scale (such as Celsius), it lacks an associated directional component (a vector direction such as $\hat{i}, \hat{j}, \hat{k}$ or an orientation in 3D space) required to qualify as a vector [By the principles of Scalar Algebra].
Visual Representation of a Scalar Scalar Representation
Step 4: Classification
| Measure | Category | Reasoning |
|---|---|---|
| $40^\circ$ | Scalar | Possesses magnitude only; lacks directional component. |
Final Solution: The measure $40^\circ$ is classified as 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(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
- 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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