Q.If either $\vec{a}=\vec{0}$ or $\vec{b}=\vec{0}$, then $\vec{a}\times\vec{b}=\vec{0}$. Is the converse true? Justify your answer with an example.

The key idea is that the cross product of two vectors is zero if and only if the vectors are parallel (or one is zero). The converse asks: if $\vec{a}\times\vec{b}=\vec{0}$, must at least one of them be the zero vector?

Reasoning:

1. The cross product $\vec{a}\times\vec{b}=\vec{0}$ when $\vec{a}$ and $\vec{b}$ are parallel (i.e., $\vec{a} = k\vec{b}$ for some scalar $k$), or when either vector is zero.
2. The converse is false because two non-zero parallel vectors also give a zero cross product. …

The cross product $\vec{a} \times \vec{b}$ is a vector whose magnitude is $|\vec{a}||\vec{b}|\sin\theta$, where $\theta$ is the angle between them. The direction is perpendicular to both $\vec{a}$ and $\vec{b}$.

When does this product become the zero vector? The magnitude is zero if either $|\vec{a}|=0$, or $|\vec{b}|=0$, or $\sin\theta = 0$. The condition $\sin\theta = 0$ means $\theta = 0^\circ$ or $180^\circ$ — that is, the vectors are parallel (or anti-parallel). So the cross product vanishes whenever the two vectors are parallel, regardless of whether they are zero or not.

The original statement says: If either vector is zero, then the cross product is zero. That's true. The converse would be: If the cross product is zero, then either vector is zero. That is false, because the cross product is also zero when the vectors are parallel and non-zero.

Let's see this with a concrete example.

1. Choose two non-zero parallel vectors.

   Take $\vec{a} = (1, 0, 0)$ and $\vec{b} = (2, 0, 0)$. Both lie along the x-axis. Neither is the zero vector.

2. Compute their cross product.

   Using the determinant formula:

3. Interpret the result. …

Method: Disproving a Converse with a Counterexample

Same reasoning pattern as any converse question: locate the other way the conclusion can arise, then exhibit one concrete case.

Steps

Step 1: State the converse being tested.

Original: 'a zero vector $\Rightarrow \vec{a}\times\vec{b}=\vec{0}$.' Converse: '$\vec{a}\times\vec{b}=\vec{0}\Rightarrow$ some vector is zero.'

Step 2: Identify the missing case. …

Common Mistakes

Mistake 1: Believing the converse is true.

Why it's wrong: $\vec{a}\times\vec{b}=\vec{0}$ also holds for non-zero parallel vectors, so it doesn't force a zero vector. Correct approach: give a parallel non-zero counterexample.

Mistake 2: Offering perpendicular vectors as the counterexample. …