Let the vectors PQ, QR, RS, ST, TU and UP represent the sides of a regular hexagon.
STATEMENT-1 : PQ

Q: Why is $\vec{RS} + \vec{ST} = \vec{RT}$? Can't we just add components?

You absolutely could use components, but in a regular hexagon, the Triangle Law is your best friend. It turns a complex sum into a single geometric vector, saving you precious time and reducing the chance of calculation errors. In JEE Advanced, recognizing these geometric shortcuts is the difference between solving a problem in two minutes versus ten.

Q: How do I know if two vectors in a hexagon are parallel without coordinates?

Look at the symmetry! In a regular hexagon, opposite sides are always parallel. For example, $\vec{PQ}$ is parallel to $\vec{ST}$. Adjacent sides, like $\vec{PQ}$ and $\vec{RS}$, are never parallel. Use this geometric intuition to bypass tedious algebra. If you can visualize the shape, you can solve the vector problem.

Q: Wait, why is $\vec{PQ} \times \vec{ST} = \vec{0}$? They point in opposite directions!

That is a classic trap! Remember, the cross product magnitude is $|\vec{A}||\vec{B}| \sin(\theta)$. If $\theta = 180^\circ$ (anti-parallel), $\sin(180^\circ) = 0$. So, the cross product is zero for both parallel and anti-parallel vectors. Don't let the direction fool you; if the lines are parallel, the cross product vanishes!

Question

Let the vectors $\vec{PQ}, \vec{QR}, \vec{RS}, \vec{ST}, \vec{TU}$ and $\vec{UP}$ represent the sides of a regular hexagon.
STATEMENT-1 : $\vec{PQ} 	imes (\vec{RS} + \vec{ST}) 
eq \vec{0}$. because
STATEMENT-2 : $\vec{PQ} 	imes \vec{RS} = \vec{0}$ and $\vec{PQ} 	imes \vec{ST} 
eq \vec{0}$.

Options:

(A) Statement-1 is True, statement-2 is True; Statement-2 is a correct explanation for Statement-1.

(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

(C) Statement-1 is True, Statement-2 is False

(D) Statement-1 is False, Statement-2 is True.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The regular hexagon $PQRSTU$ provides a symmetric playground for vector analysis. We define the sides as vectors $\vec{PQ}$, $\vec{QR}$, $\vec{RS}$, $\vec{ST}$, $\vec{TU}$, and $\vec{UP}$.

Because these vectors form a closed loop, their sum is zero:
$$\vec{PQ} + \vec{QR} + \vec{RS} + \vec{ST} + \vec{TU} + \vec{UP} = \vec{0}$$

### The Tool of the Trade: The Cross Product

Our primary analytical tool is the cross product, defined as $\vec{A} 	imes \vec{B} = |\vec{A}||\vec{B}| \sin(	heta) \hat{n}$. In the context of JEE Advanced, this serves as a **parallelism detector**.

If $\vec{A} 	imes \vec{B} = \vec{0}$, then $\sin(	heta) = 0$, implying the vectors are parallel or anti-parallel. If the cross product is non-zero, the vectors are not parallel.

### Decoding Statement-1: The Triangle Law

Statement-1 requires us to evaluate $\vec{PQ} 	imes (\vec{RS} + \vec{ST})$. By the **Triangle Law of Vector Addition**, the sum of two consecutive vectors $\vec{RS}$ and $\vec{ST}$ is simply the resultant vector $\vec{RT}$.

The expression simplifies to:
$$\vec{PQ} 	imes \vec{RT}$$

In a regular hexagon, $\vec{PQ}$ is a side and $\vec{RT}$ is a diagonal. Since these vectors are not parallel, their cross product cannot be zero. Therefore, **Statement-1 is True**.

### The Trap of Statement-2

Statement-2 makes two distinct claims. First, it asserts that $\vec{PQ} 	imes \vec{RS} = \vec{0}$.

Since $\vec{PQ}$ and $\vec{RS}$ are not parallel (the angle between them is $120^\circ$), their cross product is non-zero. Thus, the first part of Statement-2 is **False**.

Second, it claims $\vec{PQ} 	imes \vec{ST} 
eq \vec{0}$. In a regular hexagon, $\vec{PQ}$ and $\vec{ST}$ are opposite sides, meaning they are parallel (specifically, $\vec{PQ} = -\vec{ST}$).

Because they are parallel, their cross product must be zero. Since the statement claims it is non-zero, the second part of Statement-2 is also **False**.

### The Final Victory

We have navigated the geometry, applied the Triangle Law, and used the cross product as our compass. We have determined that **Statement-1 is True** and **Statement-2 is False**.

Remember, in mathematics, the answer is the result of a logical narrative. By breaking down the geometry and visualizing the vector relationships, we have successfully decoded the problem.

Animated Solution for Mathematics - Vector Algebra: Let the vectors $P Q, QR, R S, S T, T U$ and $U P$ represent the sides of a regular hexagon. STATEMENT-1 : $P Q \times (R S + S T) \neq = 0$ . because STATEMENT-2 : $P Q \times R S = 0$ and $P Q \times S T \neq = 0$ .

Select Answer:

Visualized Solution

Visualizing the Regular Hexagon

Understanding Vector Cross Product and Parallelism

Vector Addition: $R S + S T$

Evaluating Statement-1: $P Q \times R T$

Statement-1 Conclusion

Analyzing Statement-2: Part A

Analyzing Statement-2: Part B

Final Conclusion

The Sigma Insight: Vector (Cross) Product

Analyzing the Setup

The Tool of the Trade

Decoding Statement-1

The Trap of Statement-2

The Final Victory

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