Vectors

Coordinates and Components

• Definition in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$

• Vector Norm (Magnitude)

• Normalization (Unit vectors)

Dot Product (Scalar Product)

• Geometric Definition: $\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|\cos \theta$

• Algebraic Definition

• Orthogonality conditions

• Projection of a vector onto another

Cross Product (Vector Product)

• Definition via Determinant

• Geometric Interpretation (Area of parallelogram)

• Right-hand rule

• Properties ($\vec{u}\times \vec{v}=-\vec{v}\times \vec{u}$)

Mixed Product

• Scalar Triple Product

• Volume of Parallelepiped and Tetrahedron

Analytic Geometry

• Lines: Parametric and Cartesian equations

• Planes: Normal vector, Cartesian equation

• Distances:

  • Point-to-Line

  • Point-to-Plane

• Spheres and Circles

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