← Back to FractionRush

➡️ Vectors

Grade 9 · Cambridge IGCSE · Algebra & Geometry

Notation

Bold a, arrow →, column vector (x;y)

Magnitude

|v| = √(x² + y²)

Addition & Subtraction

Add/subtract x and y components

Scalar Multiplication

Multiply each component by k

Position Vectors

OA, midpoint, parallel vectors

Vector Proofs

Show paths using given vectors

1. Vector Notation

A vector has both magnitude (size) and direction. Scalars only have magnitude.

Three ways to write a vector:
Bold letter: a (printed) or a (handwritten)
Arrow notation: AB⃗ (from A to B)
Column vector: a = ⟨3, 4⟩ or written as a 2×1 matrix (3 above 4)
In exams, underline vectors when writing by hand. Never write a vector without indicating it's a vector.

2. Magnitude of a Vector

The magnitude (length) of vector v = (x, y) is:

|v| = √(x² + y²)
Example: v = (3, 4)
|v| = √(3² + 4²) = √(9 + 16) = √25 = 5

3. Adding and Subtracting Vectors

Add or subtract the x-components, then the y-components independently:

a + b = (ax + bx, ay + by)
a − b = (ax − bx, ay − by)
a = (2, 3), b = (1, −1)
a + b = (3, 2)    a − b = (1, 4)

4. Scalar Multiplication

ka = (kax, kay)
a = (2, 3):   2a = (4, 6)    −a = (−2, −3)    ½a = (1, 1.5)
If vector b = ka for some scalar k, then a and b are parallel. If k = 1, they are equal vectors.

5. Position Vectors

The position vector of point A is OA⃗, measured from the origin O.
If OA = a and OB = b, then:
AB⃗ = OB⃗ − OA⃗ = b − a
Midpoint M of AB: OM⃗ = ½(OA⃗ + OB⃗) = ½(a + b)
OA = (2, 4), OB = (6, 2)
Midpoint M: OM = ½(2+6, 4+2) = ½(8, 6) = (4, 3)

6. Parallel Vectors and Vector Proofs

Vectors are parallel if one is a scalar multiple of the other.
a = (2, 4) and b = (1, 2) are parallel because a = 2b.
Vector proof strategy:
To show AB is parallel to CD, express AB⃗ and CD⃗ in terms of known vectors. If AB⃗ = k·CD⃗, they are parallel.
In proofs, go from a known point along known vector paths, step by step: AB⃗ = AO⃗ + OB⃗. This is the "route" method.

Example 1 — Magnitude

Find |v| when v = (5, 12).

|v| = √(5² + 12²) = √(25 + 144) = √169 = 13

Example 2 — Adding Vectors

a = (3, 5) and b = (2, −1). Find a + b.

x-component: 3 + 2 = 5
y-component: 5 + (−1) = 4
a + b = (5, 4)

Example 3 — Subtracting Vectors

a = (4, 7) and b = (1, 3). Find a − b.

x: 4 − 1 = 3    y: 7 − 3 = 4
a − b = (3, 4)

Example 4 — Scalar Multiplication

a = (3, 5). Find 2a.

2a = (2×3, 2×5) = (6, 10)

Example 5 — Midpoint

OA = (2, 4), OB = (8, 6). Find the midpoint M of AB.

OM = ½(OA + OB) = ½((2+8), (4+6)) = ½(10, 10) = (5, 5)

Example 6 — Vector Path (Route Method)

O is origin, OA = a, OB = b. Find AB⃗ in terms of a and b.

Route: A → O → B
AB⃗ = AO⃗ + OB⃗ = −a + b = b − a

➡️ Vector Plotter

Enter components for vectors a and b. The plotter draws both vectors and shows a+b, a−b, and 2a.

Enter vectors and click Plot.

Exercise 1 — Magnitude |v| (to 2 d.p.)

1. v = (3, 4). Find |v|.

2. v = (5, 12). Find |v|.

3. v = (6, 8). Find |v|.

4. v = (3, 4). Find |v|. (same as Q1)

5. v = (5, 5). Find |v|.

6. v = (4, 7.5). Find |v|.

7. v = (6, 8.1). Find |v|.

8. v = (7, 11). Find |v|.

9. v = (4, 4.5). Find |v|.

10. v = (8, 15). Find |v|.

Exercise 2 — Add Vectors: x-component of a+b

1. a=(2,3), b=(3,5). x-component of a+b.

2. a=(4,1), b=(3,4). x-component of a+b.

3. a=(−3,2), b=(2,−4). x-component of a+b.

4. a=(5,−2), b=(3,13). x-component of a+b.

5. a=(1,4), b=(2,0). x-component of a+b.

6. a=(−2,3), b=(−2,−9). x-component of a+b.

7. a=(7,−1), b=(2,8). x-component of a+b.

8. a=(0,5), b=(2,−2). x-component of a+b.

9. a=(−1,2), b=(−2,1). x-component of a+b.

10. a=(4,−3), b=(2,12). x-component of a+b.

Exercise 3 — Add Vectors: y-component of a+b

1. a=(2,3), b=(3,5). y-component of a+b.

2. a=(4,1), b=(3,4). y-component of a+b.

3. a=(−3,2), b=(2,−4). y-component of a+b.

4. a=(5,6), b=(3,5). y-component of a+b.

5. a=(1,2), b=(2,2). y-component of a+b.

6. a=(−2,−3), b=(−2,−3). y-component of a+b.

7. a=(7,3), b=(2,4). y-component of a+b.

8. a=(0,1), b=(2,2). y-component of a+b.

9. a=(−1,−3), b=(−2,−2). y-component of a+b.

10. a=(4,5), b=(2,4). y-component of a+b.

Exercise 4 — Scalar Multiplication: x-component of ka

1. a=(3,2). Find x of 2a.

2. a=(3,5). Find x of 3a.

3. a=(2,1). Find x of −2a.

4. a=(4,3). Find x of 3a.

5. a=(3,4). Find x of 5a.

6. a=(2,1). Find x of −4a.

7. a=(6,2). Find x of 3a.

8. a=(2,5). Find x of 3a.

9. a=(5,3). Find x of −2a.

10. a=(7,4). Find x of 3a.

Exercise 5 — Midpoint Vector: x-component of OM

1. OA=(2,4), OB=(3,6). x-comp of midpoint M.

2. OA=(3,2), OB=(4,8). x-comp of midpoint M.

3. OA=(−2,4), OB=(1,6). x-comp of midpoint M.

4. OA=(5,2), OB=(3,4). x-comp of midpoint M.

5. OA=(0,3), OB=(2,5). x-comp of midpoint M.

6. OA=(−3,2), OB=(−1,6). x-comp of midpoint M.

7. OA=(6,1), OB=(3,7). x-comp of midpoint M.

8. OA=(1,3), OB=(2,5). x-comp of midpoint M.

9. OA=(−4,1), OB=(1,5). x-comp of midpoint M.

10. OA=(4,2), OB=(2,6). x-comp of midpoint M.

🏋️ Practice — 20 Questions

1. v=(3,4). |v|?

2. v=(5,12). |v|?

3. v=(6,8). |v|?

4. v=(3,4). |v|?

5. v=(5,5). |v| (2dp)?

6. v=(4,7.5). |v| (2dp)?

7. v=(6,8.1). |v| (2dp)?

8. v=(7,11). |v| (2dp)?

9. v=(4,4.5). |v| (2dp)?

10. v=(8,15). |v|?

11. a=(2,3), b=(3,5). x of a+b?

12. a=(4,1), b=(3,4). x of a+b?

13. a=(−3,2), b=(2,−4). x of a+b?

14. a=(5,−2), b=(3,13). x of a+b?

15. a=(1,4), b=(2,0). x of a+b?

16. a=(−2,3), b=(−2,−9). x of a+b?

17. a=(7,−1), b=(2,8). x of a+b?

18. a=(0,5), b=(2,−2). x of a+b?

19. a=(−1,2), b=(−2,1). x of a+b?

20. a=(4,−3), b=(2,12). x of a+b?

🏆 Challenge — 8 Questions

1. a=(2,3), b=(3,5). y of a+b?

2. a=(4,1), b=(3,4). y of a+b?

3. a=(−3,2), b=(2,−4). y of a+b?

4. a=(5,6), b=(3,5). y of a+b?

5. a=(1,2), b=(2,2). y of a+b?

6. a=(−2,−3), b=(−2,−3). y of a+b?

7. a=(7,3), b=(2,4). y of a+b?

8. a=(0,1), b=(2,2). y of a+b?