Algebra 1 lecture notes_V3

UNIVERSITY OF PRIMORSKA

Faculty of Mathematics, Natural Sciences and Information Technologies

ALGEBRA I Lecture Notes 2024/2025

1 VECTORS

1.1 Three-dimensional Coordinate System

• Introduction of three-dimensional space for calculus and higher mathematics.

• Building on 2D coordinate systems by adding a third axis (z-axis) for depth.

• Utilization of vectors to describe lines and planes in 3D space.

1.2 Vectors in R3

1.2.1 Component Form

• A vector in 3D is represented as a directed line segment, defined by an initial point (A) and terminal point (B).

• Definition 1.4: Vector represented by directed line segment AB has length |AB|.

• Component form: For a vector v = P Q, where P and Q in standard position, v = (x2 - x1, y2 - y1, z2 - z1).

1.2.2 Vector Algebra Operations

• Vector Addition: u + v = (u1 + v1, u2 + v2, u3 + v3).

• Scalar Multiplication: k * u = (k * u1, k * u2, k * u3).

• Geometric interpretation of vector addition via parallelogram law.

1.3 The Dot Product

• Definition 1.13: The dot product u • v = u1 * v1 + u2 * v2 + u3 * v3.

• Algebraic Properties: Additivity, homogeneity, symmetricity.

• Geometrical Meaning: u • v = |u| * |v| * cos(ϕ), ϕ is the angle between u and v.

1.4 The Cross Product

• Cross product of two vectors results in a third vector orthogonal to the plane defined by the first two.

• Definition 1.28: u × v = (|u| |v| sin(ϕ))n, where n is a unit vector orthogonal to the plane.

• Algebraic Properties: Additivity, homogeneity, anticommutativity.

1.5 The Box Product

• Definition 1.32: Box product hu × v, wi = |u × v| |w| cos(ϕ), relating to triple scalar product.

• Geometrical Meaning: The absolute value of the box product equals the volume of the parallelepiped determined by vectors u, v, w.

2 LINES AND PLANES IN R3

Lines in R3

Vector and Parametric Form

• A line in space defined by a point P0 and a direction vector v.

• Vector Form: ` = r0 + λv, λ ∈ R.

• Parametric Form: x = x0 + λv1, y = y0 + λv2, z = z0 + λv3.

Canonical Form

• Canonical equation for a line through P0 parallel to v can be expressed without zero components.

Planes in R3

• Defined by a point and a direction vector (normal vector).

• Plane through point R0, orthogonal to vector n: a(x - x0) + b(y - y0) + c(z - z0) = 0.

2.1 Intersections

• Intersecting Two Lines: Solve system for λ1 and λ2.

• Intersecting a Line and a Plane: Substitute line equation into the plane equation to find intersection point.

• Example: Determine intersection between specific line and plane equations.

2.2 Distances

• Distance from Point to Line: Use vector projection and determine point on line closest to given point.

• Distance from Point to Plane: Calculate shortest distance by moving perpendicularly from point to plane.

Additional Topics

Distance Formulas

• Distance between two points in space extended from 2D to three dimensions.

• Example problems illustrating calculations for distances between points, lines, and planes.

Conclusion

• Understanding these foundational concepts is vital for comprehending higher mathematics and applications involving geometry in three-dimensional space.