Vectors give you the tools to prove geometric results algebraically, and the edexcel igcse further pure mathematics specification tests this skill with precision
A scalar has magnitude only. A vector has both magnitude and direction. That distinction sounds simple, but the consequences fill an entire section of the specification. You need to add and subtract coplanar vectors, multiply by scalars, decompose vectors into components, find magnitudes, work with position vectors and unit vectors, prove collinearity and parallelism, and find the position vector of a point dividing a line in a given ratio. Each of these operations has a clear algebraic procedure that the exam tests directly.
These edexcel igcse further pure mathematics scalar and vector quantities revision notes cover the full section. The logical approach here is step-by-step: master the operations first, then apply them to geometric proofs. The proofs are where the marks concentrate, so spending time on the foundations is an investment that pays off in the exam.
Vector operations
Addition and subtraction
Vectors are added component by component. If a = 3i + 2j and b = -i + 5j, then:
a + b = (3 + (-1))i + (2 + 5)j = 2i + 7j.
a - b = (3 - (-1))i + (2 - 5)j = 4i - 3j.
Scalar multiplication
Multiplying a vector by a scalar scales its magnitude without changing its direction (if the scalar is positive) or reverses it (if negative).
If c = 2i - 4j, then 3c = 6i - 12j and -c = -2i + 4j.
Equal non-parallel vectors
If two non-parallel vectors are equal, then their corresponding components are equal. This is the key principle that unlocks most vector proof questions. If pa + qb = ra + sb where a and b are non-parallel, then p = r and q = s.
Components and properties of vectors
Magnitude
The magnitude of v = ai + bj is |v| = \u221a(a2 + b2).
Worked example: Find the magnitude of v = 5i - 12j.
|v| = \u221a(25 + 144) = \u221a169 = 13.
Unit vectors
A unit vector has magnitude 1. The unit vector in the direction of v is v/|v|.
Worked example: Find the unit vector in the direction of v = 3i + 4j.
|v| = \u221a(9 + 16) = 5. Unit vector = (3/5)i + (4/5)j.
Position vectors
The position vector of a point P is the vector from the origin O to P. If A has position vector a and B has position vector b, then the vector from A to B is:
AB = b - a.
Dividing a line in a given ratio
If P divides AB in the ratio m:n, then the position vector of P is:
p = (na + mb) / (m + n).
Worked example: A has position vector 2i + 3j and B has position vector 8i - j. Find the position vector of the point P that divides AB in the ratio 2:1.
p = (1(2i + 3j) + 2(8i - j)) / (2 + 1) = (2i + 3j + 16i - 2j) / 3 = (18i + j) / 3 = 6i + (1/3)j.
Geometric proofs using vectors
This is where the edexcel igcse further pure mathematics exam awards the most marks in the vectors section. The standard proof techniques are:
| Property to prove | Vector condition |
|---|---|
| Two lines are parallel | AB = kCD for some scalar k |
| Three points are collinear | AB = kAC for some scalar k (A, B, C lie on the same line) |
| Lines are concurrent | The position vectors of the intersection point satisfy all line equations simultaneously |
| A point is the midpoint of a segment | Position vector = (a + b) / 2 |
Worked example: proving collinearity
Points A, B and C have position vectors a = i + 2j, b = 4i + 5j, and c = 7i + 8j. Show that A, B and C are collinear.
AB = b - a = 3i + 3j.
AC = c - a = 6i + 6j = 2(3i + 3j) = 2AB.
Since AC = 2AB and they share the point A, the points A, B and C are collinear, with B being the midpoint of AC.
Worked example: proving a quadrilateral is a parallelogram
ABCD is a quadrilateral with position vectors a = i + j, b = 4i + 2j, c = 6i + 5j, d = 3i + 4j. Show that ABCD is a parallelogram.
AB = b - a = 3i + j.
DC = c - d = 3i + j.
Since AB = DC, AB is parallel to DC and equal in length. Therefore ABCD is a parallelogram.
Worked example: finding the ratio of division
In triangle OAB, OA = a and OB = b. M is the midpoint of OA and N is the point on AB such that AN = (1/3)AB. Find ON in terms of a and b, and determine whether M, N and B are collinear.
AB = b - a.
AN = (1/3)AB = (1/3)(b - a).
ON = a + AN = a + (1/3)(b - a) = (2/3)a + (1/3)b.
OM = (1/2)a.
MN = ON - OM = (2/3)a + (1/3)b - (1/2)a = (1/6)a + (1/3)b.
MB = b - (1/2)a = -(1/2)a + b.
Check if MB = kMN: -(1/2)a + b = k((1/6)a + (1/3)b). Comparing coefficients: k/6 = -1/2 gives k = -3, and k/3 = 1 gives k = 3. Since k = -3 and k = 3 are not equal, M, N and B are not collinear.
Practice questions
These scalar and vector quantities edexcel igcse problems cover the key exam techniques. Work through each one and check your reasoning.
- If p = 3i - 2j and q = -i + 5j, find |2p + q|.
- Find the unit vector in the direction of -4i + 3j.
- A = (1, 4), B = (7, 12). Find the position vector of the point that divides AB in the ratio 3:2.
- Show that the points P(2, 1), Q(5, 7) and R(8, 13) are collinear.
- In triangle OAB, OA = a and OB = b. P is the midpoint of OA and Q divides AB in the ratio 1:2. Express PQ in terms of a and b.
Solutions
Question 1: 2p + q = 2(3i - 2j) + (-i + 5j) = 6i - 4j - i + 5j = 5i + j. |5i + j| = \u221a(25 + 1) = \u221a26.
Question 2: |-4i + 3j| = \u221a(16 + 9) = 5. Unit vector = (-4/5)i + (3/5)j.
Question 3: p = (2a + 3b)/(3 + 2) = (2(1, 4) + 3(7, 12))/5 = ((2 + 21), (8 + 36))/5 = (23/5, 44/5). Position vector: (23/5)i + (44/5)j.
Question 4: PQ = (3, 6), PR = (6, 12) = 2(3, 6) = 2PQ. Since PR = 2PQ and they share point P, the points are collinear.
Question 5: OP = (1/2)a. AQ = (1/3)AB = (1/3)(b - a). OQ = a + (1/3)(b - a) = (2/3)a + (1/3)b. PQ = OQ - OP = (2/3)a + (1/3)b - (1/2)a = (1/6)a + (1/3)b.
Self-check questions
- Find the integral of 4x3 - 6x + 2 with respect to x.
- Evaluate the definite integral from x = 1 to x = 3 of (2x2 - x) dx. Show each step of your working.
- The velocity of a particle is given by v = 6t - t2 m/s. Find the displacement of the particle between t = 0 and t = 4 seconds.
- Find the area enclosed between the curve y = x2 - 4 and the x-axis.
- A curve has gradient function dy/dx = 3x2 - 2 and passes through the point (1, 5). Find the equation of the curve.
- A function f(x) satisfies f'(x) = 4x - 1. Given that f(2) = 7, find f(x) and evaluate f(5).
The igcse 4PM1 scalar and vector quantities section rewards rigorous algebraic working and clear geometric reasoning. These edexcel igcse further pure mathematics notes should give you the confidence to handle both computational questions and proofs. For further edexcel igcse further pure mathematics explained content and edexcel igcse further pure mathematics practice questions across every topic, use edexcel igcse further pure mathematics revision notes on the Green Bridge CBT platform to sharpen your exam readiness.
Edexcel IGCSE Further Pure Mathematics revision notes on scalar and vector quantities: operations, position vectors, unit vectors and geometric proofs.
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