Analytic geometry and linear algebra: vectors, lines, planes, matrices
What this domain tests
Vektorgeometrie and lineare Algebra in the Abitur: describing points, lines and planes in vector form, deciding their relative position (Lagebeziehung), computing angles and distances, and using matrices for transformations or for solving linear systems. Typical tasks ask for a proof (e.g. that a quadrilateral is a parallelogram), a distance, or a reflection.
Rules and concepts
- Line: x = a + t·u; plane in parameter form x = a + s·u + t·v, or in normal form (x − a)·n = 0 with n = u × v.
- Coordinate form ax + by + cz = d is the normal form multiplied out; n = (a, b, c) is a normal vector.
- Scalar product: u·v = |u||v|cos φ, so cos φ = u·v/(|u||v|); u·v = 0 tests perpendicularity.
- Distance point–plane: Hesse normal form, d = |(p − a)·n₀| with n₀ = n/|n|.
- Distance between skew lines: place a plane through one line parallel to the other (normal vector = cross product of both direction vectors), then take the point–plane distance.
- Line–plane position: substitute the line into the plane equation; one solution = intersection point, none = parallel, infinitely many = the line lies in the plane.
- Matrices: multiply, invert (2×2: swap the diagonal, change the signs off it, divide by the determinant), and read them as linear maps.
Worked example
Plane E: 2x − y + 2z = 6, so n = (2, −1, 2) and |n| = 3. Point P(4|0|1). Then (p − a)·n₀ = (2·4 − 0 + 2·1 − 6)/3 = (8 + 2 − 6)/3 = 4/3. The distance is 4/3 length units. The sign of the numerator also tells you on which side of E the point lies: here on the side the normal vector points to.
Traps
- Forgetting to normalise n - normalising is exactly what turns the formula into a distance.
- Using the direction vector of a line as the normal vector of a plane that contains it.
- Testing skew lines by comparing direction vectors only; parallel is not the same as skew, you must check whether the lines intersect.
- Losing a sign in the normal form: 2x − y + 2z = 6 has n = (2, −1, 2); writing (2, 1, 2) describes a different plane.
- Cross-product arithmetic: write each component out twice and re-check before using it.