← Further Maths syllabus

SingaporeFurther Maths

Vectors and the Geometry of Three Dimensions

5 dot points across 5 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do we find where lines and planes intersect, and how do we compute the shortest distance between geometric objects?

Find the intersection of lines and planes and compute shortest distances from a point to a line or plane and between two skew lines
A focused answer to the H2 Further Mathematics outcome on intersections and distances in 3D. The intersection of a line and a plane and of two planes, the perpendicular distance from a point to a line and to a plane, and the shortest distance between two skew lines.
12 min answer →

How do we describe a line in three dimensions, and how do we test whether two lines intersect, are parallel or are skew?

Write the vector, parametric and Cartesian equations of a line in three dimensions and classify the relationship between two lines
A focused answer to the H2 Further Mathematics outcome on lines in 3D. The vector, parametric and Cartesian forms of a line, the angle between lines, and classifying two lines as intersecting, parallel or skew.
11 min answer →

How do we describe a plane in three dimensions using a normal vector, and how do we move between its forms?

Write the vector, scalar product and Cartesian equations of a plane using a normal vector and find the angle between planes and between a line and a plane
A focused answer to the H2 Further Mathematics outcome on planes in 3D. The normal vector, the vector, scalar product and Cartesian equations of a plane, finding a normal from two directions, and the angle between two planes and between a line and a plane.
11 min answer →

What do the scalar and vector products compute, and how do we use them to find angles, areas and volumes?

Use the scalar and vector products and the scalar triple product to find angles, areas and volumes in three dimensions
A focused answer to the H2 Further Mathematics outcome on vector products. The scalar (dot) product for angles and projections, the vector (cross) product for perpendiculars and areas, and the scalar triple product for volumes and coplanarity.
11 min answer →

How do we use vector methods to prove geometric results and find reflections, foot of perpendicular and other constructions?

Apply vector methods to geometric problems including the foot of the perpendicular, reflections of points, and proofs of geometric properties
A focused answer to the H2 Further Mathematics outcome on applying vectors to geometry. Finding the foot of the perpendicular from a point to a line or plane, reflecting a point in a line or plane, and using position vectors and the ratio theorem to prove geometric results.
11 min answer →