LEC8

• Complex Vectors.
• Cross products.
• Lagrange’s Identity.

Complex Vector

• Quite literally, vector with complex numbers in them (not in syllabus)

$$C^n=\left\{\left[\begin{array}{c}{z}_1\\ .\\ .\\ z_n\end{array}\right]\right|z_1,...,z_n\in \mathbb{C}\}$$

Cross Product in $R^3$

• Take determinant.
• Returns a vector.
• The cross product is perpendicular to both the involved vectors.
• Only in $R^3$.
• Geometrically, it represents the area of the parallelogram that the two vectors form.

Lagrange’s Identity

$$\begin{array}{r}||\vec{x}\times \vec{y}||=||\vec{x}|{|}^2||\vec{y}|{|}^2-(\vec{x}\cdot \vec{y}{)}^2\\ =||\vec{x}|{|}^2||\vec{y}|{|}^2-||\vec{x}|{|}^2||\vec{y}|{|}^2{\cos }^2\theta \\ =||\vec{x}|{|}^2||\vec{y}|{|}^2(1-{\cos }^2\theta )\\ =||\vec{x}|{|}^2||\vec{y}|{|}^2{\sin }^2\theta \\ \\ ⟹||\vec{x}\times \vec{y}||=||\vec{x}||||\vec{y}||\sin \theta \end{array}$$