Let P, Q, R and S be the points on the plane with position vectors 2i - j, 4i, 3i + 3j and i + 2j re

Q: Wait, why do we calculate $\vec{SR}$ as $\vec{R} - \vec{S}$ and not $\vec{S} - \vec{R}$? Does the order matter?

Absolutely! In vector geometry, direction is everything. $\vec{SR}$ represents the vector starting at $S$ and ending at $R$. If you reverse the order, you get the negative vector, which points in the opposite direction. For a quadrilateral $PQRS$, the sides must be traversed in order: $P \to Q$, $Q \to R$, $R \to S$, and $S \to P$. Thus, the opposite side to $\vec{PQ}$ is indeed $\vec{SR}$.

Q: Why can't we just use the distance formula for everything? Why use vectors?

You absolutely can! The distance formula is just the magnitude of the vector. However, vectors are the 'superpower' of coordinate geometry. They allow us to check for parallelism (by comparing components) and perpendicularity (via the dot product) simultaneously, saving you time and reducing the chance of calculation errors.

Q: If the dot product is not zero, does that mean it's definitely not a rectangle?

Yes, exactly. The dot product of two vectors $\vec{A}$ and $\vec{B}$ is defined as $|\vec{A}||\vec{B}| \cos \theta$. If the dot product is zero, $\cos \theta$ must be zero, which implies $\theta = 90^\circ$. If the dot product is non-zero, the angle cannot be $90^\circ$, and therefore, the parallelogram cannot be a rectangle.

Question

Let $P, Q, R$ and $S$ be the points on the plane with position vectors $2\hat{i} - \hat{j}, 4\hat{i}, 3\hat{i} + 3\hat{j}$ and $\hat{i} + 2\hat{j}$ respectively. The quadrilateral $PQRS$ must be a

Options:

(A) parallelogram, which is neither a rhombus nor a rectangle

(B) square

(C) rectangle, but not a square

(D) rhombus, but not a square

Sigma Percentile · Accepted Answer

### Analyzing the Setup

To determine the nature of the quadrilateral $PQRS$ with vertices $P(2, -1)$, $Q(4, 0)$, $R(3, 3)$, and $S(1, 2)$, we must analyze the vectors representing its sides.

First, we calculate the vector for side $\vec{PQ}$:
$$\vec{PQ} = (4-2)\hat{i} + (0 - (-1))\hat{j} = 2\hat{i} + \hat{j}$$

Next, we calculate the vector for the opposite side, $\vec{SR}$:
$$\vec{SR} = \vec{R} - \vec{S} = (3-1)\hat{i} + (3-2)\hat{j} = 2\hat{i} + \hat{j}$$

Since $\vec{PQ} = \vec{SR}$, the opposite sides are equal in magnitude and parallel. This confirms that the quadrilateral is a **parallelogram**.

### The Rhombus Test

A rhombus is a special parallelogram where all sides are equal in length. We calculate the magnitudes of adjacent sides $\vec{PQ}$ and $\vec{QR}$ to check for this property.

The magnitude of $\vec{PQ}$ is:
$$|\vec{PQ}| = \sqrt{2^2 + 1^2} = \sqrt{5}$$

The vector $\vec{QR}$ is calculated as:
$$\vec{QR} = \vec{R} - \vec{Q} = (3-4)\hat{i} + (3-0)\hat{j} = -\hat{i} + 3\hat{j}$$

The magnitude of $\vec{QR}$ is:
$$|\vec{QR}| = \sqrt{(-1)^2 + 3^2} = \sqrt{10}$$

Because $\sqrt{5} 
eq \sqrt{10}$, the adjacent sides are not equal. Therefore, the shape is **not a rhombus**.

### The Rectangle Test

A rectangle is a parallelogram where adjacent sides are perpendicular, implying their dot product must be zero. We evaluate the dot product of $\vec{PQ}$ and $\vec{QR}$:

$$\vec{PQ} \cdot \vec{QR} = (2\hat{i} + \hat{j}) \cdot (-\hat{i} + 3\hat{j})$$
$$\vec{PQ} \cdot \vec{QR} = (2)(-1) + (1)(3) = -2 + 3 = 1$$

Since the dot product is $1$ and not $0$, the adjacent sides are not perpendicular. Consequently, the shape is **not a rectangle**.

### Final Verdict

We have systematically evaluated the geometric properties of the quadrilateral.

1. It is a **parallelogram** because $\vec{PQ} = \vec{SR}$.
2. It is **not a rhombus** because the side lengths are unequal.
3. It is **not a rectangle** because the dot product of adjacent sides is non-zero.

The quadrilateral $PQRS$ is a **general parallelogram**.

Animated Solution for Mathematics - Vector Algebra: Let $P, Q, R$ and $S$ be the points on the plane with position vectors $2 \hat{i} - \hat{j}, 4 \hat{i}, 3 \hat{i} + 3 \hat{j}$ and $\hat{i} + 2 \hat{j}$ respectively. The quadrilateral $P QR S$ must be a

Select Answer:

Visualized Solution

Visualizing Points $P, Q, R, S$

Finding Vector $P Q$

Finding Vector $S R$

Parallelogram Condition

Checking for Rhombus: Length of $P Q$

Checking for Rhombus: Length of $QR$

Checking for Rectangle: Dot Product

Evaluating the Dot Product

Final Conclusion

The Sigma Insight: Scalar (Dot) Product

Analyzing the Setup

The Rhombus Test

The Rectangle Test

Final Verdict

Similar Questions

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Select Answer:

Visualized Solution

Visualizing Points .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P,Q,R,S

Finding Vector .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } PQ​

Finding Vector .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } SR