What vector topics are in the DSE M2 syllabus?

Three learning units: introduction to vectors (operations, i j k form), scalar product and vector product, and applications of vectors (parallelism, orthogonality, division of a line segment, projection, angle between vectors, area of a triangle or parallelogram). Equations of lines and planes are not in the syllabus.

How do I find the magnitude of a 3D vector?

Square each component, add them and take the square root: |xi + yj + zk| = √(x² + y² + z²). For example |2i − 4j + 4k| = √36 = 6.

Are direction cosines required in DSE M2?

No. The curriculum notes state that the concept of direction cosines is not required. You only need the angle a 2D vector makes with the positive x-axis.

Vectors in DSE M2: i, j, k Form, Magnitude and Operations

A vector has both magnitude and direction. In DSE M2 a position vector is written OP = xi + yj + zk, its magnitude is |OP| = √(x² + y² + z²), and the vector from A to B is AB = OB − OA.

Solve Introduction to Vectors questions with AI

The vectors OA, OB and AB = OB − OA from Example 1, drawn in 3D. Try it yourself →

Vector Formulas (DSE M2)

Vector between two pointsAB = OB − OA = b − a

Magnitude|xi + yj + zk| = √(x² + y² + z²)

Unit vectorâ = a / |a|

Parallel vectorsa = λb for some real number λ

Mid-point of ABOM = (a + b) / 2

Comparing coefficientsIf a, b are non-zero and not parallel and λa + μb = λ₁a + μ₁b, then λ = λ₁ and μ = μ₁

How to Express a Vector in Terms of a and b

Mark the origin O and write the given position vectors: OA = a, OB = b.
Find a route from the start point to the end point along lines you already know, for example OP = OA + AP.
Turn each ratio into a fraction of a known vector: if AP : PB = 1 : 2 then AP = (1/3)AB = (1/3)(b − a).
Collect the a terms and the b terms.
If two different routes give the same vector, compare the coefficients of a and b to solve for the unknown ratios.

Worked Examples — Introduction to Vectors

Example 1: Vector between two points, magnitude and unit vector

Let OA = 2i + j − 2k and OB = 4i − 3j + 2k. Find AB, |AB| and the unit vector in the direction of AB.

AB = OB − OA = (4 − 2)i + (−3 − 1)j + (2 − (−2))k = 2i − 4j + 4k
|AB| = √(2² + (−4)² + 4²) = √36 = 6
Unit vector = AB / |AB| = (1/3)i − (2/3)j + (2/3)k

Answer: AB = 2i − 4j + 4k, |AB| = 6, (1/3)i − (2/3)j + (2/3)k

Example 2: Expressing a vector in terms of a and b

In triangle OAB, OA = a and OB = b. M is the mid-point of AB and G lies on OM with OG : GM = 2 : 1. Express OG in terms of a and b.

OM = (a + b) / 2
OG = (2/3) OM
OG = (2/3) · (a + b) / 2 = (a + b) / 3

Answer: OG = (a + b) / 3

💡 Exam Traps & Secrets

⚠️ AB is "end minus start"AB = OB − OA, not OA − OB. Writing it the wrong way round flips every sign and usually costs the method mark for everything that follows.

⚠️ Comparing coefficients needs non-parallel vectorsYou may equate the coefficients of a and b only when they are non-zero and not parallel. State this before you compare, or the step is not justified.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Vector between two points)Given OA = i + 2j + 3k and OB = 4i − 2j + k, find AB and |AB|.

(Unit vector)Find the unit vector in the direction of 3i − 4j + 12k.

(Section formula)In triangle OAB, P divides AB in the ratio 1 : 3. Express OP in terms of a and b.