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Q5(ii):
Answer the following as true or false.
(ii) Two collinear vectors are always equal in magnitude.
Solution :
Analytical Definition of Collinear Vectors
Two vectors $\vec{a}$ and $\vec{b}$ are defined as collinear (or parallel) if there exists a non-zero scalar $\lambda$ such that:
$$\vec{b} = \lambda \vec{a}$$
[Per the definition of collinearity in Vector Algebra, vectors are collinear if they are parallel to the same line, irrespective of their magnitudes or directions.]
Step 1: Evaluating the Relationship Between Collinearity and Magnitude
Consider the magnitudes of the two vectors. If $\vec{b} = \lambda \vec{a}$, then by the property of vector magnitude:
$$|\vec{b}| = |\lambda \vec{a}| = |\lambda| |\vec{a}|$$
The condition for the two vectors to be equal in magnitude is $|\vec{b}| = |\vec{a}|$, which implies $|\lambda| = 1$. This only occurs if $\lambda = 1$ or $\lambda = -1$.
Step 2: Counter-Example Construction
To determine if the statement "Two collinear vectors are always equal in magnitude" is true, we test for a counter-example where $\lambda \neq \pm 1$.
- Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$.
- Let $\lambda = 2$.
- Define $\vec{b} = \lambda \vec{a} = 2\hat{i} + 2\hat{j} + 2\hat{k}$.
Calculations:
| Vector | Magnitude Formula | Result |
|---|---|---|
| $|\vec{a}|$ | $\sqrt{1^2 + 1^2 + 1^2}$ | $\sqrt{3}$ |
| $|\vec{b}|$ | $\sqrt{2^2 + 2^2 + 2^2}$ | $\sqrt{12} = 2\sqrt{3}$ |
Since $\sqrt{3} \neq 2\sqrt{3}$, the vectors are collinear but their magnitudes are not equal. [By the property of scalar multiplication of vectors].
Visual Representation of Collinear Vectors with Different Magnitudes
Conclusion
The statement is False because collinear vectors are defined by parallelism (the existence of a scalar multiple), not by identity of magnitude. Only if the scalar multiple $\lambda$ is restricted to $\pm 1$ would the magnitudes be equal.
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(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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