Page 4: Collinearity and Vector Operations

This page focuses on determining collinearity of points and performing various vector operations.

Key topics: • Testing for collinearity using vector equations • Solving parametric equations to find specific points • Vector addition and scalar multiplication

Vocabulary: Collinear points lie on the same straight line.

Example: To check if three points are collinear, their position vectors are used to form an equation: A + t(B-A) = C, where t is a scalar parameter.

Highlight: The page emphasizes the importance of verifying solutions by substituting results back into original equations.

Page 5: Intersection of Lines and Planes

This page covers the intersection of lines with planes and verifying calculated results.

Key concepts: • Determining the intersection point of a line and a plane • Using parametric equations to solve for intersection points • Verifying results through substitution

Example: The intersection of a line a: x = (3, 1, 2) + t(3, 2, 1) with a plane is found by solving for the parameter t and then substituting back into the line equation.

Highlight: The page stresses the importance of checking solutions by substituting the calculated intersection point into both the line and plane equations.

Page 1: Introduction and Problem Setup

This page introduces a set of vector geometry problems for a mathematics exam. The questions cover various aspects of 3D vector calculations and geometric concepts.

Key points: • A triangle ABC is given with vertices A(4|5|3), B(-1|3|-4), and C(3|-1|-2) • Students must plot the triangle, find a fourth point to form a parallelogram, and reflect point A • A parametric equation for a line parallel to the x₁-axis through point B is required • The concept of the centroid (Schwerpunkt) of a triangle is introduced

Definition: The centroid of a triangle is the intersection point of its medians, which connect each vertex to the midpoint of the opposite side.

Highlight: The centroid divides each median in a 2:1 ratio, with the longer segment closer to the vertex.