Q.Area of a rectangle having vertices A, B, C and D with position vectors $-\hat i+\dfrac{1}{2}\hat j+4\hat k$, $\hat i+\dfrac{1}{2}\hat j+4\hat k$, $\hat i-\dfrac{1}{2}\hat j+4\hat k$ and $-\hat i-\dfrac{1}{2}\hat j+4\hat k$, respectively is (A) $\dfrac{1}{2}$ (B) $1$ (C) $2$ (D) $4$

Concept: Vector Magnitude Properties — the area of a rectangle equals the product of the lengths of two adjacent sides.

Step 1: Pick two adjacent sides.

$\overrightarrow{AB} = (1 - (-1))\hat i + \left(\frac12 - \frac12\right)\hat j + (4-4)\hat k = 2\hat i$

$\overrightarrow{AD} = (-1 - (-1))\hat i + \left(-\frac12 - \frac12\right)\hat j + (4-4)\hat k = -\hat j$

Step 2: Find their lengths. …

The four vertices are given as position vectors. The first thing to notice is that all four have the same $z$-coordinate: $4$. That means the entire rectangle lies in a plane parallel to the $xy$-plane. So we can ignore the $z$-component entirely — the rectangle’s shape is determined only by the $x$ and $y$ coordinates.

Let’s list the vertices in order:

• $A = ( -1,\; \frac12,\; 4 )$
• $B = ( 1,\; \frac12,\; 4 )$
• $C = ( 1,\; -\frac12,\; 4 )$
• $D = ( -1,\; -\frac12,\; 4 )$

The problem says these are vertices of a rectangle in that order. Check: $A$ to $B$ changes only $x$ (from $-1$ to $1$), $B$ to $C$ changes only $y$ (from $\frac12$ to $-\frac12$), $C$ to $D$ changes only $x$ (back to $-1$), and $D$ to $A$ changes only $y$ (back to $\frac12$). So the sides are aligned with the axes — a clean axis-aligned rectangle.

Now the area of a rectangle is length $\times$ breadth. We just need the side lengths.

1. Side $AB$ (horizontal): $x$ goes from $-1$ to $1$, so length = $|1 - (-1)| = 2$. The $y$ and $z$ are constant, so this is the full length. …

Method: Area of a Rectangle from Vertex Position Vectors

When a rectangle's vertices are given as position vectors, its area is the product of the lengths of two adjacent sides — no angle needed.

Steps

Step 1: Form two adjacent side vectors

List the vertices in order $A,B,C,D$. Take two sides sharing a vertex:

subtracting component-wise (head minus tail).

Step 2: Find each side length

Compute $|\overrightarrow{AB}|$ and $|\overrightarrow{AD}|$ using $|\vec v|=\sqrt{x^2+y^2+z^2}$.

Step 3: Multiply (or cross-check with the cross product) …

Common Mistakes

Mistake 1: Multiplying a side length by a diagonal length

Why it's wrong: choosing $\overrightarrow{AC}$ (a diagonal) instead of an adjacent side inflates the area. Correct approach: use two sides meeting at one vertex, e.g. $\overrightarrow{AB}$ and $\overrightarrow{AD}$, never $\overrightarrow{AC}$.

Mistake 2: Dragging the $z$-coordinate into the length even though it is constant …