1) Definition

The vector product (or cross product) of two vectors \(\vec{A}\) and \(\vec{B}\) is defined as

\[
\boxed{\;\vec{A}\times\vec{B} \;=\; |\vec{A}|\,|\vec{B}|\,\sin\theta \;\hat{n}\;}
\]

where
• \(\theta\) is the angle between \(\vec{A}\) and \(\vec{B}\) (\(0^\circ\leq\theta\leq 180^\circ\)),
• \(\hat{n}\) is a unit vector perpendicular to both \(\vec{A}\) and \(\vec{B}\), in the direction given by the right-hand rule.
The result is a vector.

2) Geometrical Meaning

The magnitude of the cross product represents the area of the parallelogram formed by \(\vec{A}\) and \(\vec{B}\):

\(|\vec{A}\times\vec{B}| = |\vec{A}||\vec{B}|\sin\theta\).

Hence, the area of the triangle formed by vectors \(\vec{A}\) and \(\vec{B}\) is \(\tfrac{1}{2}|\vec{A}\times\vec{B}|\).

3) Right-Hand Rule

The direction of \(\vec{A}\times\vec{B}\) is perpendicular to the plane containing \(\vec{A}\) and \(\vec{B}\).
Using the Right-Hand Rule:

• Point your right-hand fingers from \(\vec{A}\) to \(\vec{B}\) through the smaller angle.
• Your thumb points in the direction of \(\vec{A}\times\vec{B}\).

Note: \(\vec{B}\times\vec{A} = -(\vec{A}\times\vec{B})\).

4) Determinant Form (Component Formula)

If \(\vec{A}=A_x\hat{i}+A_y\hat{j}+A_z\hat{k}\) and \(\vec{B}=B_x\hat{i}+B_y\hat{j}+B_z\hat{k}\), then

\[
\vec{A}\times\vec{B} =
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
A_x & A_y & A_z \\
B_x & B_y & B_z
\end{vmatrix}
\]

Expanding,

\(\vec{A}\times\vec{B} = (A_yB_z – A_zB_y)\hat{i} – (A_xB_z – A_zB_x)\hat{j} + (A_xB_y – A_yB_x)\hat{k}\).

5) Properties

• Anticommutative: \(\vec{A}\times\vec{B}=-(\vec{B}\times\vec{A})\).
• Self product: \(\vec{A}\times\vec{A}=0\).
• Distributive: \(\vec{A}\times(\vec{B}+\vec{C}) = \vec{A}\times\vec{B} + \vec{A}\times\vec{C}\).
• Scalar multiplication: \((\lambda\vec{A})\times\vec{B}=\lambda(\vec{A}\times\vec{B})\).
• Magnitude: \(|\vec{A}\times\vec{B}|=|\vec{A}||\vec{B}|\sin\theta\).
• Direction: Perpendicular to both vectors (right-hand rule).

6) Angle Between Vectors

From cross product magnitude:

\(\sin\theta = \dfrac{|\vec{A}\times\vec{B}|}{|\vec{A}||\vec{B}|}\).

Useful when dot product (cos θ) gives complementary information.

7) Applications of Vector Product

• Area of Parallelogram/Triangle: \(|\vec{A}\times\vec{B}|\), \(\tfrac{1}{2}|\vec{A}\times\vec{B}|\).
• Torque: \(\vec{\tau} = \vec{r}\times\vec{F}\).
• Angular Momentum: \(\vec{L} = \vec{r}\times\vec{p}\).
• Magnetic Force on Charge: \(\vec{F} = q(\vec{v}\times\vec{B})\).
• Normal vector to a plane: For vectors along edges, cross product gives perpendicular normal.

8) Solved Examples

9) Practice Questions

1. \(\vec{A}=\langle 1,2,3\rangle,\;\vec{B}=\langle 4,5,6\rangle.\) Find \(\vec{A}\times\vec{B}\).
2. If \(|\vec{A}|=3,\;|\vec{B}|=4,\;\theta=30^\circ\), find \(|\vec{A}\times\vec{B}|\).
3. Find the area of the parallelogram with adjacent sides \(\vec{a}=\langle 2,1,0\rangle,\;\vec{b}=\langle 1,1,1\rangle\).
4. Two forces \(\vec{F}_1=\langle 3,2,0\rangle,\;\vec{F}_2=\langle 1,0,4\rangle\) act at a point. Find torque about origin if position vector is \(\vec{r}=\langle 0,1,2\rangle\).
5. Show that \(\vec{i}\times\vec{j}=\vec{k},\;\vec{j}\times\vec{k}=\vec{i},\;\vec{k}\times\vec{i}=\vec{j}\).

10) Answer Key (Practice)

1. \(\vec{A}\times\vec{B}=\langle -3,6,-3\rangle\).
2. \(|\vec{A}\times\vec{B}|=|\vec{A}||\vec{B}|\sin 30^\circ=3\cdot4\cdot\tfrac12=6\).
3. \(|\vec{a}\times\vec{b}|=\sqrt{(1\cdot1-0\cdot1)^2-(2\cdot1-0\cdot1)^2+(2\cdot1-1\cdot1)^2}=\sqrt{1+(-2)^2+(1)^2}=\sqrt{6}.\)
4. Torque=\(\vec{r}\times(\vec{F}_1+\vec{F}_2)=\langle 0,1,2\rangle\times\langle 4,2,4\rangle=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\0&1&2\\4&2&4\end{vmatrix}= (1\cdot4-2\cdot2)\hat{i}-(0\cdot4-2\cdot4)\hat{j}+(0\cdot2-1\cdot4)\hat{k}=(0)\hat{i}-( -8)\hat{j}+(-4)\hat{k}=8\hat{j}-4\hat{k}.\)
5. By cyclic rule of unit vectors: \(\hat{i}\times\hat{j}=\hat{k}\), etc.