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Sec 4 Vectors | Position Vectors, Magnitude, Translation and Vector Geometry

Secondary 4 Vectors is the eduKateSingapore guide to two-dimensional vectors: directed line segments, position vectors, vector addition and subtraction, scalar multiplication, magnitude, translations and geometric proofs. For students searching vectors O Level, Sec 4 vectors, position vectors, vector geometry or column vectors, the main shift is to stop treating vectors as arrows drawn on a page and start reading them as movements with both magnitude and direction.

The 2027 G3 SEC Mathematics syllabus K310 keeps two-dimensional vectors as a full geometry subtopic: vector notation, directed line segments, translations, position vectors, magnitude, sums and differences, scalar multiplication and geometric problems. This page is therefore written as a durable SEC-ready owner while remaining useful to the 2026 4052 cohort.

This page sits inside the Secondary Mathematics Topic Library. It depends on coordinate reasoning, algebra and geometry, and it stays within G3/E-Math scope. Deeper vector methods, including more advanced vector geometry, belong in the specialist Mathematics handoff to Bukit Timah Tutor.

Quick answer: what is a vector?

A vector describes a displacement with both magnitude and direction.

A scalar has magnitude only. Examples of scalars include mass, time and temperature. A vector such as displacement needs direction as well as size.

Column-vector notation

A vector can be written as a column such as (3, −2), meaning 3 units horizontally and −2 units vertically.

The exact display may be typeset vertically in examination notation, but the component meaning is unchanged.

Directed line segments

Vector AB means the displacement from A to B. Vector BA points in the opposite direction.

BA = −AB.

The order of letters matters.

Position vectors

If O is the origin, vector OA is the position vector of point A.

If A has coordinates (4,−1), its position vector from O is represented by those same components.

Vector addition

Adding vectors means performing one displacement after another.

If a=(2,3) and b=(4,−1), then a+b=(6,2).

Graphically, place the tail of b at the head of a. The resultant runs from the original start to the final endpoint.

Vector subtraction

a−b = a + (−b).

If a=(5,4) and b=(2,7), then a−b=(3,−3).

Subtraction is addition of the opposite vector.

Scalar multiplication

Multiplying vector a by scalar k changes its magnitude by |k| and may reverse direction if k is negative.

  • 2a: same direction, twice the magnitude.
  • ½a: same direction, half the magnitude.
  • −a: same magnitude, opposite direction.

Magnitude of a vector

For vector (x,y), magnitude = √(x²+y²).

This is Pythagoras on the horizontal and vertical components.

Worked magnitude example

Vector v=(6,8).

|v|=√(36+64)=10.

Translation by a vector

A translation moves every point by the same vector.

If point P=(2,5) is translated by vector (3,−4), the image is P′=(5,1).

Translation preserves shape, size and orientation.

Finding AB from position vectors

If OA=a and OB=b, then AB=b−a.

This relation is one of the most useful vector identities in coordinate geometry.

Worked example

A=(2,1), B=(7,5).

AB=(7−2,5−1)=(5,4).

Magnitude AB=√(25+16)=√41.

Expressing vectors in terms of two base vectors

Suppose OA=a and OB=b. If point M is the midpoint of AB, then:

OM = ½(OA+OB)=½(a+b).

This is a model of averaging two position vectors.

Midpoint by vectors

If M is midpoint of AB, AM=MB.

Since AB=b−a, AM=½(b−a).

Then OM=a+½(b−a)=½(a+b).

Ratio division

If P divides AB in a known ratio, vectors can express OP as a weighted combination of OA and OB.

At G3 level, the exact structure should be derived from the section ratio rather than memorised without meaning.

Worked internal division example

A point P lies one-third of the way from A to B.

AP=⅓AB=⅓(b−a).

OP=OA+AP=a+⅓(b−a)=⅔a+⅓b.

Parallel vectors

Two non-zero vectors are parallel if one is a scalar multiple of the other.

If PQ=3AB, then PQ is parallel to AB and points in the same direction.

If PQ=−2AB, the lines are parallel but the vector direction is opposite.

Collinearity

Points A, B and C are collinear if vectors AB and AC are scalar multiples of each other.

This converts a geometric statement into an algebraic vector test.

Worked collinearity example

A=(1,2), B=(3,6), C=(4,8).

AB=(2,4). AC=(3,6).

AC=1.5AB, so A, B and C are collinear.

Vector proofs in geometry

A vector proof usually aims to show one of four structures:

  • two sides are parallel
  • two sides are equal as vectors
  • a point lies on a line
  • a point is a midpoint or divides a line in a fixed ratio

The proof becomes clear when every vector is expressed in the same base vectors.

Worked parallelogram structure

Let OA=a, OB=b and OC=a+b.

Then AC=(a+b)−a=b, while OB=b.

Thus AC=OB as vectors.

Similarly BC=(a+b)−b=a=OA.

Opposite sides are equal and parallel, supporting parallelogram structure.

Vectors versus coordinates

Coordinates locate points; vectors describe displacements between points. The two systems work together.

A coordinate difference is a vector; a position vector behaves like the coordinates of the endpoint relative to the origin.

Common vector mistakes

  • reverses AB and BA
  • adds coordinates instead of subtracting to find AB
  • treats vectors as ordinary lengths
  • forgets direction when using negative scalars
  • mixes points and vectors in notation
  • claims parallelism without showing scalar multiple
  • uses magnitude when the problem needs the full vector

The vector audit

  1. What are the base points and directions?
  2. Am I working with a point, a position vector or a displacement vector?
  3. If finding AB, did I use B−A?
  4. Can every vector be expressed in the same basis?
  5. Is the final claim about equality, parallelism, magnitude or position?
  6. Does the sign match the direction?

Practice

Practice 1

Question: a=(3,4): |a|

Answer: 5

Practice 2

Question: A=(2,1), B=(5,7): AB

Answer: (3,6)

Practice 3

Question: a=(2,−1), b=(4,3): a+b

Answer: (6,2)

Practice 4

Question: a=(2,−1): −3a

Answer: (−6,3)

Practice 5

Question: OA=a, OB=b: midpoint M of AB

Answer: OM=½(a+b)

Practice 6

Question: AC=2AB

Answer: A,B,C are collinear if directions are consistent

2027 SEC transition note

K310 G3 Mathematics explicitly retains vectors in two dimensions, including notation, position vectors, magnitude, translation, scalar multiplication and geometric applications. The examination name changes; the vector language does not.

Frequently asked questions

What is the difference between a vector and a scalar?

A vector has magnitude and direction; a scalar has magnitude only.

How do I find vector AB?

Position vector of B minus position vector of A.

How do I prove two lines are parallel with vectors?

Show one direction vector is a scalar multiple of the other.

What does multiplying by a negative scalar do?

It reverses direction and scales magnitude.

Where does vector magnitude come from?

Pythagoras applied to the vector components.

Where does this sit in Atlas?

This is the canonical Sec 4 Vectors owner under Secondary Mathematics Topic Library.

The final Vectors rule

Vectors are geometry written as movement. Keep the direction visible, express every displacement in a common basis, and use scalar multiples to turn visual claims about lines and points into algebraic proof.

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