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Q5(iv):
Answer the following as true or false.
(iv) Two collinear vectors having the same magnitude are equal.
Solution :
Step 1: Defining the Properties of Vectors
To analyze the statement, we must formally define the criteria for vector equality. Two vectors $\mathbf{a}$ and $\mathbf{b}$ are considered equal if and only if:
- They have the same magnitude ($|\mathbf{a}| = |\mathbf{b}|$).
- They have the same direction.
[Per the definition of Equality of Vectors in Vector Algebra].
Step 2: Analyzing Collinearity
Two vectors $\mathbf{a}$ and $\mathbf{b}$ are collinear if they are parallel to the same line, irrespective of their magnitudes and directions. Mathematically, $\mathbf{b} = k\mathbf{a}$ for some non-zero scalar $k$. [By the definition of Collinear Vectors].
If two vectors are collinear and have the same magnitude, there are two distinct possibilities for their direction:
- Case A: They point in the same direction (i.e., $k = 1$). In this case, the vectors are indeed equal.
- Case B: They point in opposite directions (i.e., $k = -1$). In this case, the vectors are negatives of each other ($\mathbf{a} = -\mathbf{b}$), not equal.
Step 3: Logical Counter-Example
Consider two vectors $\mathbf{a}$ and $\mathbf{b}$ such that $\mathbf{a} = \hat{i}$ and $\mathbf{b} = -\hat{i}$.
- Magnitude: $|\mathbf{a}| = \sqrt{1^2} = 1$ and $|\mathbf{b}| = \sqrt{(-1)^2} = 1$. Thus, $|\mathbf{a}| = |\mathbf{b}|$. [By Euclidean Norm definition].
- Collinearity: $\mathbf{b} = -1 \cdot \mathbf{a}$. Since they satisfy the scalar multiplication condition, they are collinear.
- Equality: Since the direction of $\mathbf{a}$ (positive x-axis) is opposite to the direction of $\mathbf{b}$ (negative x-axis), $\mathbf{a} \neq \mathbf{b}$.
Step 4: Conclusion
Because there exists a subset of collinear vectors with the same magnitude that possess opposite directions (making them unequal), the universal statement "Two collinear vectors having the same magnitude are equal" is logically invalid.
Final Solution: False
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(iii): Answer the following as true or false. (iii) Two vectors having same magnitude are collinear.
CBSE Solutions for Class 12 Mathematics Vector Algebra
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