Q.Given that $\vec{a}\cdot\vec{b}=0$ and $\vec{a}\times\vec{b}=\vec{0}$. What can you conclude about the vectors $\vec{a}$ and $\vec{b}$?

Concept: Orthogonal And Parallel

Since $\vec{a}\cdot\vec{b}=0$, the vectors are perpendicular (orthogonal).

Since $\vec{a}\times\vec{b}=\vec{0}$, the vectors are parallel (collinear). …

When you see both a dot product and a cross product given as zero, it looks contradictory at first glance. The dot product being zero tells you the vectors are orthogonal (perpendicular). The cross product being zero tells you they are parallel (or one is zero). How can two vectors be both perpendicular and parallel at the same time?

The answer lies in the zero vector. The zero vector is special — it is considered both orthogonal to every vector (since $\vec{0} \cdot \vec{b} = 0$ for any $\vec{b}$) and parallel to every vector (since $\vec{0} \times \vec{b} = \vec{0}$ for any $\vec{b}$). So the only way both conditions can be true is if at least one of the vectors is the zero vector.

Let's walk through the reasoning step by step.

1. What the dot product tells us.

   The dot product $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta = 0$.

   This means either $|\vec{a}| = 0$, or $|\vec{b}| = 0$, or $\cos \theta = 0$ (i.e., $\theta = 90^\circ$). So the vectors are either perpendicular, or one of them is the zero vector.

2. What the cross product tells us.

   The magnitude of the cross product is $|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta = 0$.

   This means either $|\vec{a}| = 0$, or $|\vec{b}| = 0$, or $\sin \theta = 0$ (i.e., $\theta = 0^\circ$ or $180^\circ$). So the vectors are either parallel (or anti-parallel), or one of them is the zero vector.

3. Combining both conditions. …

Method: Reconciling Simultaneous Dot = 0 and Cross = 0

This is a reasoning question: two conditions that look contradictory are resolved by the zero vector.

Steps

Step 1: Translate each condition.

$\vec{a}\cdot\vec{b}=0$ (carrying $\cos\theta$) says perpendicular OR a zero vector; $\vec{a}\times\vec{b}=\vec{0}$ (carrying $\sin\theta$) says parallel OR a zero vector.

Step 2: Check whether a non-zero angle can satisfy both. …

Common Mistakes

Mistake 1: Declaring the conditions contradictory and giving up.

Why it's wrong: two non-zero vectors can't be both perpendicular and parallel, but the zero vector resolves it. Correct approach: conclude at least one vector is $\vec{0}$.

Mistake 2: Concluding the vectors are simply perpendicular (or simply parallel).

Why it's wrong: that uses only one of the two given conditions. Correct approach: both $\cos\theta=0$ and $\sin\theta=0$ can't hold for a real angle, so a magnitude must be zero. …