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Q4(ii):
In Fig 10.6 (a square), identify the following vectors. (ii) Equal
In Fig 10.6 (a square), identify the following vectors. (ii) Equal
Solution :
Given Variables & Initial Setup
Consider a square $ABCD$ in a 2D plane. Based on the geometric properties of a square, the vectors representing the sides and diagonals are defined by their direction and magnitude. In a square $ABCD$, let the vertices be represented as vectors from the origin or as displacement vectors between vertices.
Theoretical Definition of Equal Vectors
Two vectors $\vec{a}$ and $\vec{b}$ are defined as equal if and only if:
- They have the same magnitude ($|\vec{a}| = |\vec{b}|$).
- They have the same direction (parallel and pointing in the same orientation).
[Per the standard definition of vector equality in Euclidean space].
Step 1: Analyzing the Vectors of the Square
In square $ABCD$, the sides can be represented as follows:
- $\vec{AB}$ and $\vec{DC}$: These vectors are parallel and equal in length. Since the square has parallel opposite sides ($AB \parallel DC$), $\vec{AB} = \vec{DC}$ because they possess identical magnitude (length of the side) and direction.
- $\vec{AD}$ and $\vec{BC}$: These vectors are parallel and equal in length. Since $AD \parallel BC$, $\vec{AD} = \vec{BC}$ because they possess identical magnitude and direction.
Step 2: Verification of Equality
| Vector Pair | Magnitude Equal? | Direction Same? | Equal? |
|---|---|---|---|
| $\vec{AB}, \vec{DC}$ | Yes ($a = a$) | Yes | Yes |
| $\vec{AD}, \vec{BC}$ | Yes ($a = a$) | Yes | Yes |
| $\vec{AB}, \vec{AD}$ | Yes | No (orthogonal) | No |
Step 3: Synthesis of Result
Based on the geometric properties of the square $ABCD$ provided in the coordinate plane:
1. The vector $\vec{AB}$ is equal to the vector $\vec{DC}$ as they are directed along the same axis (parallel) and have the same magnitude $a$.
2. The vector $\vec{AD}$ is equal to the vector $\vec{BC}$ as they are directed along the same axis (parallel) and have the same magnitude $a$.
Final Solution: The equal vectors in the square $ABCD$ are $\vec{AB} = \vec{DC}$ and $\vec{AD} = \vec{BC}$.
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(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.
- 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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