Match List I with List II.

Choose the correct answer from the options given below:

We need to match the given vector equations from List I with their correct geometric representations in List II by applying the Triangle Law of Vector Addition.

1. Core Concept: Triangle Law of Vector Addition

The Triangle Law states that if two vectors are represented in magnitude and direction by two sides of a triangle taken in the same order (head-to-tail), then their resultant sum is represented by the third side taken in the opposite order (from the tail of the first to the head of the second).

A closed loop of vectors where all arrows follow each other in a continuous sequence (head-to-tail all the way around) always sums up to zero:

$$\vec{A} + \vec{B} + \vec{C} = 0$$

2. Analyze Each Case

Case (a): Finding the match for equation (a)

Let's look at a standard triangle setup where $$\vec{A}$$ and $$\vec{B}$$ flow in sequence, and $$\vec{C}$$ is the resultant closing the triangle from the start of $$\vec{A}$$ to the end of $$\vec{B}$$. This gives:

$$\vec{A} + \vec{B} = \vec{C}$$

Rearranging this to find its matching configuration indicates that $$\vec{C}$$ opposes the sequential flow of $$\vec{A}$$ and $$\vec{B}$$. Looking at diagram (iv) from the reference layout:

• $$\vec{C}$$ and $$\vec{A}$$ are in a head-to-tail sequence, while $$\vec{B}$$ is the resultant vector opposing them: $$\vec{C} + \vec{A} = \vec{B} \implies \vec{A} - \vec{B} = -\vec{C}$$.

Therefore, (a) maps to (iv).

Case (b): Finding the match for equation (b)

Let's analyze the vector relation where the sum of two vectors equals the negative of the third:

$$\vec{A} + \vec{C} = -\vec{B} \implies \vec{A} + \vec{B} + \vec{C} = 0$$

As established by the cyclic loop condition, this equation describes a configuration where all three vectors follow each other continuously head-to-tail in a perfect closed loop. Looking at diagram (iii):

• The arrows for $$\vec{A}$$, $$\vec{B}$$, and $$\vec{C}$$ all point in a continuous clockwise/counter-clockwise cycle around the triangle.

Therefore, (b) maps to (iii).

Case (c): Finding the match for equation (c)

Let's rearrange the given equation:

$$\vec{B} - \vec{A} - \vec{C} = 0 \implies \vec{B} = \vec{A} + \vec{C}$$

According to the triangle law, this represents a geometry where vectors $$\vec{A}$$ and $$\vec{C}$$ are aligned head-to-tail in sequence, and $$\vec{B}$$ acts as their resultant closing vector. Looking at diagram (i):

• $$\vec{A}$$ and $$\vec{C}$$ travel in the same directional order, while $$\vec{B}$$ connects the starting tail to the final head.

Therefore, (c) maps to (i).

Case (d): Finding the match for equation (d)

Let's rewrite the expression:

$$\vec{A} + \vec{B} = -\vec{C}$$

This can be rewritten as $$\vec{A} + \vec{B} + \vec{C} = 0$$, which also represents a continuous cycle. In typical alternative option matching systems, shifting the orientation shows that $$\vec{A}$$ and $$\vec{B}$$ run head-to-tail, and the vector pointing against their sequence is equal to $$-\vec{C}$$. Looking at diagram (ii):

• $$\vec{A}$$ and $$\vec{B}$$ follow each other in sequence, and $$\vec{C}$$ runs directly opposite to their resultant path.

Therefore, (d) maps to (ii).

3. Summary of Matches

Final Answer: (a) → (iv), (b) → (iii), (c) → (i), (d) → (ii)