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Q4(iii):

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].

A(0,a) B(a,a) C(a,0) D(0,0)

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 $ABCD$:

  1. 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.
  2. Consider vector $\vec{AD}$ and vector $\vec{BC}$. Both lie on vertical lines ($x=0$ and $x=a$ respectively).
  3. 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


CBSE Solutions for Class 12 Mathematics Vector Algebra


Chapters in CBSE - Class 12 Mathematics


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