Cross Product of Vectors
Vector Algebra: Vector Product Formulas
1. Definition & Calculation
| Method | Formula |
|---|---|
| Definition ($$ \hat{n} $$ is unit normal vector) |
$$ \vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin\theta \, \hat{n} $$ |
| Determinant Form (Coordinate Method) |
$$ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} $$ |
| Sine of Angle (Magnitude only) |
$$ \sin\theta = \frac{|\vec{a} \times \vec{b}|}{|\vec{a}| |\vec{b}|} $$ |
| Unit Normal Vector ($$ \perp $$ to both $$ \vec{a} $$ and $$ \vec{b} $$) |
$$ \hat{n} = \frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|} $$ |
2. Key Properties
Parallelism ($$ \parallel $$)
If $$ \vec{a} \parallel \vec{b} $$ ($$ \theta = 0 $$ or $$ \pi $$):
$$ \vec{a} \times \vec{b} = \vec{0} $$
If $$ \vec{a} \parallel \vec{b} $$ ($$ \theta = 0 $$ or $$ \pi $$):
$$ \vec{a} \times \vec{b} = \vec{0} $$
Self Product
$$ \vec{a} \times \vec{a} = \vec{0} $$
$$ \vec{a} \times \vec{a} = \vec{0} $$
Anti-Commutative
$$ \vec{a} \times \vec{b} = – (\vec{b} \times \vec{a}) $$
$$ \vec{a} \times \vec{b} = – (\vec{b} \times \vec{a}) $$
| Standard Unit Vectors | $$ \hat{i} \times \hat{j} = \hat{k}, \quad \hat{j} \times \hat{k} = \hat{i}, \quad \hat{k} \times \hat{i} = \hat{j} $$ (Reverse order gives negative: $$ \hat{j} \times \hat{i} = -\hat{k} $$) |
| Distributive Property | $$ \vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c} $$ |
3. Geometric Applications (Area)
| Shape | Area Formula |
|---|---|
| Area of Triangle (Adjacent sides $$ \vec{a}, \vec{b} $$) |
$$ \text{Area} = \frac{1}{2} |\vec{a} \times \vec{b}| $$ |
| Area of Parallelogram (Adjacent sides $$ \vec{a}, \vec{b} $$) |
$$ \text{Area} = |\vec{a} \times \vec{b}| $$ |
| Area of Parallelogram (Given Diagonals $$ \vec{d_1}, \vec{d_2} $$) |
$$ \text{Area} = \frac{1}{2} |\vec{d_1} \times \vec{d_2}| $$ |