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Q5(iii):
Answer the following as true or false.
(iii) Two vectors having same magnitude are collinear.
Solution :
Theoretical Foundation & Definitions
To evaluate the statement "Two vectors having same magnitude are collinear," we must strictly define the geometric and algebraic properties of vectors in $\mathbb{R}^3$:
- Magnitude of a Vector: The length of a vector $\vec{a}$, denoted as $|\vec{a}|$, is a scalar value representing its displacement magnitude.
- Collinear Vectors: Two vectors $\vec{a}$ and $\vec{b}$ are collinear (or parallel) if and only if there exists a non-zero scalar $k$ such that $\vec{a} = k\vec{b}$. [Per the definition of parallel/collinear vectors in Vector Algebra].
Step 1: Constructing a Counter-Example
Consider two vectors in a 2D Cartesian plane:
Let $\vec{a} = 1\hat{i} + 0\hat{j}$
Let $\vec{b} = 0\hat{i} + 1\hat{j}$
Step 2: Analysis of Magnitudes
Calculate the magnitude of $\vec{a}$:
$|\vec{a}| = \sqrt{1^2 + 0^2} = 1$ [By the Euclidean norm formula]
Calculate the magnitude of $\vec{b}$:
$|\vec{b}| = \sqrt{0^2 + 1^2} = 1$ [By the Euclidean norm formula]
Observation: $|\vec{a}| = |\vec{b}|$. The magnitudes are identical.
Step 3: Testing for Collinearity
For $\vec{a}$ and $\vec{b}$ to be collinear, there must exist a scalar $k$ such that $\vec{a} = k\vec{b}$:
$1\hat{i} + 0\hat{j} = k(0\hat{i} + 1\hat{j})$
$1\hat{i} + 0\hat{j} = 0\hat{i} + k\hat{j}$
Equating components:
- $1 = 0$ (Contradiction)
- $0 = k$
Since the component equality fails, no such scalar $k$ exists. Therefore, $\vec{a}$ and $\vec{b}$ are not collinear. [By the Principle of Vector Equivalence].
Visual Representation
Logical Conclusion
Since we have demonstrated a case where vectors possess equal magnitude but are not parallel (they are orthogonal in the provided example), the universal claim that equality of magnitude implies collinearity is false.
Final Solution: False. Two vectors having the same magnitude are not necessarily collinear. Collinearity requires the vectors to be scalar multiples of each other, which magnitude equality does not guarantee.
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
- 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(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
Chapters in CBSE - Class 12 Mathematics
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