Q.State True or False: If $\vec{a}$ and $\vec{b}$ are adjacent sides of a rhombus, then $\vec{a}\cdot\vec{b}=0$.

Concept: Vector Magnitude Properties — in a rhombus, all sides are equal, so $|\vec{a}| = |\vec{b}|$, but the dot product depends on the angle between them, not just the side lengths.

Step 1: For a rhombus, adjacent sides have equal length: $|\vec{a}| = |\vec{b}|$.

Step 2: The dot product is $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = |\vec{a}|^2 \cos\theta$. …

The statement tries to connect two separate ideas: the shape of a rhombus and the condition for perpendicular vectors. A rhombus is defined as a quadrilateral with all four sides equal. That’s it. There is no requirement that its angles be $90^\circ$.

The dot product $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$ equals zero only when $\cos \theta = 0$, i.e., when $\theta = 90^\circ$. So the statement is claiming that any rhombus must have perpendicular adjacent sides — which is not true. Only a square (a special rhombus) satisfies that.

Let’s walk through the reasoning step by step.

1. Recall the definition of a rhombus

   A rhombus is a parallelogram with all four sides of equal length. If $\vec{a}$ and $\vec{b}$ are adjacent sides, then $|\vec{a}| = |\vec{b}|$. That’s the only condition guaranteed.

2. What the dot product tells us

   The dot product $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$, where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$. For this to be zero, we need $\cos \theta = 0$, i.e., $\theta = 90^\circ$.

3. Does a rhombus force $\theta = 90^\circ$?

   No. Consider a rhombus that is not a square — for example, a diamond shape with acute and obtuse angles. The adjacent sides are still equal in length, but the angle between them is not $90^\circ$. So $\vec{a} \cdot \vec{b} \neq 0$ in general. …

Method: Separating a Shape's Defining Property from an Extra Assumption

Use this for true/false claims that attach a perpendicularity (or angle) conclusion to a shape defined only by its side lengths.

Steps

Step 1: State exactly what the shape guarantees

A rhombus is defined by all sides equal, so adjacent sides satisfy $|\vec{a}|=|\vec{b}|$ — and nothing about the angle between them.

Step 2: Write the dot product in angle form …

Common Mistakes

Mistake 1: Confusing a rhombus with a square

Why it's wrong: a rhombus guarantees only equal sides, not right angles; only a square (a special rhombus) has $\vec{a}\cdot\vec{b}=0$. Correct approach: adjacent sides are perpendicular only in the square case.

Mistake 2: Assuming equal-length sides make $\vec{a}\cdot\vec{b}=0$ …