Addition, subtraction and scalar multiplication of vectors

Q: Do I need to use i,j component notation for this topic?

No, i,j component notation is covered separately in S7T02. This topic uses bold/underlined vector notation and geometric representations only, per the 4PM1 spec.

Q: Can I apply the coefficient equating rule to all pairs of vectors?

No, the rule only works for non-parallel vectors. Parallel vectors can be written as scalar multiples of each other, so infinitely many coefficient solutions exist.

Edexcel International GCSE Further Pure Mathematics· Section S7, Specification 7A· 20 min read

1. Vector Addition & Subtraction (Geometric Rules)★★☆☆☆⏱ 4 min

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📘 Definition

Triangle Law of Vector Addition

To add two vectors a and b, place the tail of b at the head of a. The resultant vector a + b runs from the tail of a to the head of b.

Example:

If you walk 3m east (a) then 4m north (b), your resultant displacement is a + b = 5m north-east.

📐 Worked Example

Given vectors p (2 units horizontal right) and q (1 unit vertical up), find the resultant p + q using the triangle law.

1
Draw p as a horizontal arrow 2 units long, pointing right.
2
Place the tail of q at the head of p, drawing q as a 1 unit long vertical arrow pointing up.
3
Draw a line from the tail of p to the head of q: this is the resultant p + q.

Vector subtraction follows the same rule as addition, but you reverse the direction of the vector being subtracted: $a - b = a + (- b)$ .

📐 Worked Example

Calculate p - q using the same vectors p and q from the previous example.

1
Reverse the direction of q to get -q, a 1 unit vertical arrow pointing down.
2
Place the tail of -q at the head of p.
3
Draw the resultant from the tail of p to the head of -q to get p - q.

Exam tip:

Always label vectors with arrows in your diagrams to avoid direction errors, which are a common marking deduction.

2. Scalar Multiplication of Vectors★★☆☆☆⏱ 4 min

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📘 Definition

Scalar Multiplication

Multiplying a vector v by a scalar $k$ produces a new vector with magnitude $∣ k ∣ \times ∣ v ∣$ . Its direction matches v if $k > 0$ , is opposite to v if $k < 0$ , and becomes the zero vector if $k = 0$ .

Example:

If v is 3 units right, $2 v$ is 6 units right, $- 0.5 v$ is 1.5 units left.

📐 Worked Example

Given vector r = 2 units pointing up, find $3 r$ and $- 2 r$ .

1
For $3 r$ : Multiply the magnitude of r by 3: $2 \times 3 = 6$ units. Keep direction the same as r (up).
2
For $- 2 r$ : Multiply the magnitude of r by 2: $2 \times 2 = 4$ units. Reverse the direction of r to point down.

3. Equating Coefficients of Non-Parallel Vectors★★★☆☆⏱ 6 min

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📘 Definition

Coefficient Equating Rule

If $α_{1} a + β_{1} b = α_{2} a + β_{2} b$ where a and b are non-parallel coplanar vectors, you can equate coefficients of matching vectors: $α_{1} = α_{2}$ and $β_{1} = β_{2}$ .

Example:

If $2 a + 5 b = x a + (y + 1) b$ , then $x = 2$ and $y = 4$ .

📐 Worked Example

Find constants $p$ and $q$ such that $3 (p a - 2 b) = 12 a + q b$ , where a and b are non-parallel.

1
First expand the left-hand side of the equation: $3 p a - 6 b = 12 a + q b$ .
2
Equate coefficients of a: $3 p = 12$ , so $p = 4$ .
3
Equate coefficients of b: $- 6 = q$ , so $q = - 6$ .

Exam tip:

Never assume vectors are non-parallel unless explicitly stated in the question; the coefficient equating rule does not apply to parallel vectors.

4. Combined Vector Operations★★★☆☆⏱ 6 min

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You will often need to combine addition, subtraction, and scalar multiplication in one problem. Follow standard order of operations: complete scalar multiplication first, then expand brackets, then group and combine like vectors.

📐 Worked Example

Simplify $2 (a + 3 b) - 4 (2 a - b)$ , then find values of $m$ and $n$ if the simplified expression equals $m a + n b$ (a, b non-parallel).

1
Expand all scalar multiplications first, distributing negative signs correctly: $2 a + 6 b - 8 a + 4 b$ .
2
Group like vectors: $(2 a - 8 a) + (6 b + 4 b)$ .
3
Simplify to get $- 6 a + 10 b$ .
4
Equate coefficients: $m = - 6$ and $n = 10$ .

✓ Quick check

Simplify $3 (x - 2 y) + 5 (y - x)$
- $- 2 x - y$
- $2 x + y$
- $- 2 x + 11 y$

5. Common Pitfalls

Wrong move:

Forgetting to reverse the direction of the vector being subtracted

Why:

Vector subtraction is addition of the negative vector, so direction reversal is required to get the correct resultant

Correct move:

When calculating $a - b$ , first flip the direction of b to get -b, then add to a

Wrong move:

Applying the coefficient equating rule to parallel vectors

Why:

Parallel vectors can be written as scalar multiples of each other, so infinitely many coefficient solutions exist

Correct move:

Only equate coefficients if the question explicitly states the vectors are non-parallel

Wrong move:

Failing to distribute negative signs when expanding scalar multiplication

Why:

For example, expanding $- 2 (a - b)$ as $- 2 a - 2 b$ instead of $- 2 a + 2 b$ leads to wrong coefficient values

Correct move:

Distribute the scalar including its sign to every term inside brackets before combining like vectors

Wrong move:

Omitting arrow labels on vector diagrams

Why:

Examiners cannot verify the direction of your resultant vector if no arrow is present, leading to lost marks

Correct move:

Always draw an arrow on every vector you plot, including resultant vectors, to indicate direction

Wrong move:

Reversing the direction of a vector when multiplying by a positive scalar

Why:

Positive scalars only change the magnitude of a vector, not its direction

Correct move:

Keep direction identical to the original vector for positive scalars; reverse direction only for negative scalars

6. Quick Reference Cheatsheet

Operation	Rule	Example
Vector Addition (Triangle Law)	Place head of first vector at tail of second; resultant from tail of first to head of second	$a + b$ = resultant of sequential displacements
Vector Subtraction	$a - b = a + (- b)$ : reverse direction of subtracted vector then add	$a - a = 0$ (zero vector)
Scalar Multiplication	$k v$ has magnitude $∣ k ∣ \times ∣ v ∣$ , direction same if $k > 0$ , opposite if $k < 0$	$3 v$ is 3x length of $v$ , same direction
Coefficient Equating	For non-parallel $a, b$ : $α_{1} a + β_{1} b = α_{2} a + β_{2} b ⟹ α_{1} = α_{2}, β_{1} = β_{2}$	$2 a + 3 b = x a + y b ⟹ x = 2, y = 3$

7. Frequently Asked

Do I need to use i,j component notation for this topic?

No, i,j component notation is covered separately in S7_T02. This topic uses bold/underlined vector notation and geometric representations only, per the 4PM1 spec.

Can I apply the coefficient equating rule to all pairs of vectors?

No, the rule only works for non-parallel vectors. Parallel vectors can be written as scalar multiples of each other, so infinitely many coefficient solutions exist.

Going deeper

official_specEdexcel 4PM1 SpecificationRefer to section S7 for vector content

What's Next

Now that you have mastered core vector operations, you are ready to progress to more advanced vector topics in the Edexcel IGCSE Further Pure Maths syllabus. The next step is learning to represent vectors in component (i,j) form, which lets you perform operations numerically rather than geometrically, making complex problems much faster to solve. You will then build on this to calculate vector magnitudes, work with position vectors, use unit vectors, and eventually apply vector operations to geometric proofs. All these topics are regularly tested in 4PM1 exams, so solidifying your understanding of the core operations covered here is critical for success in later vector questions, which usually make up 5-8 marks of the written paper. Practice combining operations and equating coefficients regularly to avoid common errors in your exam.