Define vector product. Explain the properties of a vector product with two examples.

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> [!TLDR]
> The vector product of two vectors gives a new vector perpendicular to both, with magnitude |a||b|sinθ; it is anti-commutative and its magnitude equals the parallelogram area formed by the two vectors.
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> **Definition:** The vector product (or cross product) of two vectors $\vec{a}$ and $\vec{b}$, inclined at angle $	heta$ to each other, is defined as:
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> $\vec{a} 	imes \vec{b} = |\vec{a}||\vec{b}|\sin	heta \, \hat{n}$
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> where $\hat{n}$ is a unit vector perpendicular to the plane containing $\vec{a}$ and $\vec{b}$, with its direction given by the **right-hand rule**: curl the fingers of the right hand from $\vec{a}$ towards $\vec{b}$ through the smaller angle; the thumb then points along $\hat{n}$.
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> **Properties:**
> 1. **Not commutative (anti-commutative):** $\vec{a}	imes\vec{b} = -(\vec{b}	imes\vec{a})$ — reversing the order reverses the direction of the resultant vector.
> 2. **Distributive over addition:** $\vec{a}	imes(\vec{b}+\vec{c}) = \vec{a}	imes\vec{b} + \vec{a}	imes\vec{c}$.
> 3. **Magnitude = area of parallelogram:** $|\vec{a}	imes\vec{b}|$ equals the area of the parallelogram formed with $\vec{a}$ and $\vec{b}$ as adjacent sides.
> 4. **Cross product of a vector with itself is zero:** $\vec{a}	imes\vec{a} = 0$, since $	heta = 0 \Rightarrow \sin	heta = 0$. More generally, the cross product of two parallel (or antiparallel) vectors is zero. …

Q.Define vector product. Explain the properties of a vector product with two examples.

Concept understanding — Scalars & Vectors

Q.Define vector product. Explain the properties of a vector product with two examples.

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