National 5 Maths: Vectors in 2D and 3D

Vectors at National 5 go beyond adding two column vectors in the plane. The course asks you to add and subtract vectors as directed line segments, to read the coordinates of a point from a three-dimensional object, to add and subtract three-dimensional vectors using components and to find the magnitude of a vector in two or three dimensions. This material trains those geometric skills and complements the category's existing practice on two-dimensional components.

The quiz has twelve questions. Four use directed line segments: the diagonals of a parallelogram, the third side of a triangle taken in the reverse direction, and the vector to the midpoint of a side. Two ask for three-dimensional coordinates, one on a cuboid with a vertex at the origin and one at the apex of a square-based pyramid; each object is described fully in words, so no diagram is needed. Two work with components, $\mathbf{a}+\mathbf{b}$ and $2\mathbf{a} - \mathbf{b}$ for column vectors in three dimensions. Three find a magnitude, including the vector between two points, and the last finds an unknown component from a given magnitude. The wrong options come from the usual slips: reversing a direction, forgetting to halve for a midpoint, mixing up the order of the coordinates, and adding components instead of squaring them.

The flashcards explain what a directed line segment is, how to add vectors head to tail, what the negative of a vector is, how components describe a vector in three dimensions, the magnitude formulae in two and three dimensions, how to find the vector between two points, and what the resultant of several vectors is.

The printable written work has eight longer questions to answer by hand: vector paths in a regular hexagon and in a parallelogram, the coordinates of every vertex of a described cuboid and the length of its space diagonal, magnitudes of a sum compared with the sum of magnitudes, the resultant of three forces, the vector and midpoint between two points, a combination of two-dimensional vectors, and an unknown component where you explain why the positive value is chosen. Each asks for working and short reasons.

In the oral practice exam an examiner asks one question at a time, describes every shape in words, asks you to say which path you follow, and ends with brief feedback. The material suits S4 pupils preparing for National 5 and gives a good base for the vectors of Higher Mathematics.

• Express vectors in a shape as sums and differences of given directed line segments
• Find the vector to the midpoint of a side
• State the coordinates of a point on a three-dimensional object described in words
• Add, subtract and multiply three-dimensional vectors using components
• Calculate the magnitude of a vector in two or three dimensions, including the vector between two points
• Find an unknown component from a given magnitude

Practice material written by Zestly, based on the National 5 Mathematics course specification (Qualifications Scotland, formerly SQA; version 3.0, May 2023), geometric skills: two-dimensional vectors as directed line segments, three-dimensional coordinates, vector components and magnitude.