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Q4(iii):
In Fig 10.6 (a square), identify the following vectors. (iii) Collinear but not equal

In Fig 10.6 (a square), identify the following vectors. (iii) Collinear but not equal

Solution :
Step 1: Geometric Definition and Coordinate Assignment
To identify vectors that are collinear but not equal in a square $ABCD$, we establish a Cartesian coordinate system. Let the vertices of the square be defined as $A(0, a)$, $B(a, a)$, $C(a, 0)$, and $D(0, 0)$, where $a > 0$ represents the side length of the square. [By definition of a square in a 2D plane].
Step 2: Defining Collinear and Equal Vectors
- Collinear Vectors: Two vectors $\vec{p}$ and $\vec{q}$ are collinear if they are parallel to the same line, irrespective of their magnitudes and directions. [Per the definition of collinearity in vector space $\mathbb{R}^n$].
- Equal Vectors: Two vectors are equal if they have the same magnitude and the same direction.
Step 3: Analytical Identification
We evaluate pairs of parallel sides of the square $ABCD$:
- Consider vector $\vec{AB}$ and vector $\vec{DC}$. Both lie on horizontal lines ($y=a$ and $y=0$ respectively). Thus, they are collinear. However, their directions are identical, and if their lengths are equal, they are equal vectors.
- Consider vector $\vec{AD}$ and vector $\vec{BC}$. Both lie on vertical lines ($x=0$ and $x=a$ respectively).
- To satisfy "Collinear but not equal," we look for vectors that lie on the same or parallel lines but differ in magnitude or direction.
Step 4: Logical Deduction for the Specific Question
In the standard square configuration provided in Figure 10.6:
- Vector $\vec{a} = \vec{AB}$ and vector $\vec{c} = \vec{DC}$ are parallel. Because they have the same direction and magnitude, $\vec{AB} = \vec{DC}$. Therefore, they are not "not equal."
- If we define vectors based on the sides, $\vec{AD}$ and $\vec{BC}$ have the same magnitude. $\vec{AD}$ points from $(0,a)$ to $(0,0)$, and $\vec{BC}$ points from $(a,a)$ to $(a,0)$. Thus $\vec{AD} = \vec{BC}$.
- The solution set: Vectors $\vec{a}$ and $\vec{c}$ (where $\vec{a} = \vec{AB}$ and $\vec{c} = \vec{CD}$) are collinear because they are parallel. However, since $\vec{CD} = -\vec{AB}$, they have opposite directions. Therefore, $\vec{AB} \neq \vec{CD}$.
Final Solution: The vectors $\vec{AB}$ and $\vec{CD}$ are collinear because they are parallel to the same line, but they are not equal because they point in opposite directions ($\vec{AB} = -\vec{CD}$).
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
- 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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