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Q5(i):
Answer the following as true or false.
(i) $\vec{a}$ and $-\vec{a}$ are collinear.
Solution :
Step 1: Definition of Collinear Vectors
Two vectors $\vec{a}$ and $\vec{b}$ are defined as collinear if and only if they are parallel to the same line, irrespective of their magnitudes and directions. Mathematically, two non-zero vectors $\vec{a}$ and $\vec{b}$ are collinear if there exists a non-zero scalar $\lambda$ such that:
$\vec{b} = \lambda \vec{a}$
[Per the definition of parallel/collinear vectors in Euclidean Space]
Step 2: Analysis of the Given Vectors
We are given two vectors, $\vec{a}$ and $-\vec{a}$. We seek to determine if there exists a scalar $\lambda$ that satisfies the collinearity condition.
Let $\vec{b} = -\vec{a}$. We can rewrite this expression as:
$\vec{b} = (-1) \vec{a}$
Here, the scalar $\lambda = -1$. Since $\lambda \neq 0$, the condition for collinearity is satisfied.
Step 3: Geometric Interpretation
Geometrically, the vector $-\vec{a}$ has the same magnitude as $\vec{a}$ but acts in the exact opposite direction. Because both vectors lie along the same line of support (the line passing through the origin and the terminal point of $\vec{a}$), they are inherently collinear.
Step 4: Logical Conclusion
- Since $\vec{b} = \lambda \vec{a}$ holds for $\lambda = -1$, the vectors are parallel.
- Parallel vectors, by definition, share the same support line or are parallel to the same line.
- Therefore, $\vec{a}$ and $-\vec{a}$ are collinear.
Final Solution: True. $\vec{a}$ and $-\vec{a}$ are collinear because $-\vec{a} = (-1)\vec{a}$, satisfying the condition for collinearity.
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(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
Chapters in CBSE - Class 12 Mathematics
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