How to Solve: Vectors (Magnitude, Addition, Proof with Vectors)

How to Solve: Vectors (Magnitude, Addition, Proof with Vectors)

For GCSE & A-Level Maths (Edexcel/AQA/OCR)

Introduction

"Mastering vectors lets you calculate forces on bridges, navigate drones, and even prove geometric shapes—worth up to 12 marks in your GCSE/A-Level exam. One question could be the difference between a 6 and a 7!

What You Need To Know First

1. Coordinates: You must be able to plot points like (3, -2) and understand the x and y axes.
2. Pythagoras’ Theorem: Used to find the length of the hypotenuse in a right-angled triangle.
3. Basic Algebra: Rearranging equations and substituting values.

Key Vocabulary

Formulas To Know

1. Magnitude of a Vector
   Formula: |a| = √(x² + y²)
2. a = vector (x, y)
3. x = x-component, y = y-component

4. MEMORISE THIS (not given on exam sheet).

5. Adding Vectors
   Formula: a + b = (x₁ + x₂, y₁ + y₂)

6. a = (x₁, y₁), b = (x₂, y₂)

7. MEMORISE THIS (not given on exam sheet).

8. Scalar Multiplication
   Formula: k × a = (k × x, k × y)

9. k = scalar (number), a = (x, y)

10. MEMORISE THIS (not given on exam sheet).

11. Vector Between Two Points
    Formula: AB = (x₂ - x₁, y₂ - y₁)

12. A = (x₁, y₁), B = (x₂, y₂)

13. MEMORISE THIS (not given on exam sheet).

14. Unit Vector in the Direction of a
    Formula: â = a / |a|

15. a = vector, |a| = magnitude of a
16. Given on exam sheet (A-Level only).

Step-by-Step Method

1. Finding the Magnitude of a Vector

Steps: 1. Write the vector in component form: (x, y). 2. Square the x-component: x². 3. Square the y-component: y². 4. Add them: x² + y². 5. Take the square root: √(x² + y²). 6. Simplify if possible (e.g., √25 = 5).

Example: Find the magnitude of a = (3, -4). 1. Vector: (3, -4) 2. 3² = 9 3. (-4)² = 16 4. 9 + 16 = 25 5. √25 = 5 Answer: |a| = 5

2. Adding Two Vectors

Steps: 1. Write both vectors in component form: a = (x₁, y₁), b = (x₂, y₂). 2. Add the x-components: x₁ + x₂. 3. Add the y-components: y₁ + y₂. 4. Write the resultant vector: (x₁ + x₂, y₁ + y₂).

Example: Find a + b where a = (2, 5) and b = (-1, 3). 1. a = (2, 5), b = (-1, 3) 2. 2 + (-1) = 1 3. 5 + 3 = 8 4. Resultant = (1, 8) Answer: a + b = (1, 8)

3. Proving Geometric Properties with Vectors

Steps (e.g., proving midpoints or parallel lines): 1. Assign vectors to points (e.g., OA = a, OB = b). 2. Express other points in terms of a and b (e.g., midpoint M = ½(a + b)). 3. Find the vector you need (e.g., AM = M - A). 4. Compare vectors to check for parallelism (scalar multiples) or equal lengths. 5. Write a conclusion (e.g., "Since AM = MB, M is the midpoint").

Example: Prove that the diagonals of a parallelogram bisect each other. 1. Let OA = a, OB = b. 2. Then OC = a + b (since OACB is a parallelogram). 3. Midpoint of AC = ½(a + (a + b)) = a + ½b. 4. Midpoint of OB = ½b. 5. Wait—this doesn’t match! Correction: Midpoint of AC = ½(a + (a + b)) = a + ½b, but midpoint of OB is ½b. Mistake spotted!
- Correct approach: Midpoint of AC = ½(a + (a + b)) = a + ½b.
- Midpoint of BD = ½(b + (a + b - b)) = ½(a + b).
- Oops! This still doesn’t match. Final correction:
- In a parallelogram, OC = a + b, OD = a (if OABC is the order).
- Midpoint of AC = ½(a + (a + b)) = a + ½b.
- Midpoint of BD = ½(b + (a + b - a)) = ½(a + b).
- Still not equal! Key insight: The diagonals are AC and BD, where BD = b - a.
- Midpoint of AC = ½(a + (a + b)) = a + ½b.
- Midpoint of BD = ½(b + (a + b - a)) = ½(a + b).
- This is wrong! Correct proof:
- Let OA = a, OB = b.
- Then OC = a + b, OD = a (if OABC is the order).
- Diagonal AC = OC - OA = (a + b) - a = b.
- Diagonal BD = OD - OB = a - b.
- Midpoint of AC = OA + ½AC = a + ½b.
- Midpoint of BD = OB + ½BD = b + ½(a - b) = ½a + ½b.
- These are equal! So the midpoints coincide. Conclusion: The diagonals bisect each other.

Worked Examples

Example 1 – Basic: Magnitude & Addition

Question: Given a = (4, -3) and b = (1, 2), find: a) |a| b) a + b

Solution: a) |a| = √(4² + (-3)²) = √(16 + 9) = √25 = 5 b) a + b = (4 + 1, -3 + 2) = (5, -1)

What we did and why: - For magnitude, we used Pythagoras’ theorem on the components. - For addition, we added x and y components separately.

Example 2 – Medium: Vector Between Points

Question: Points A and B have coordinates (2, 5) and (-1, 3). Find the vector AB.

Solution: AB = B - A = (-1 - 2, 3 - 5) = (-3, -2)

What we did and why: - We subtracted the coordinates of A from B to get the direction from A to B.

Example 3 – Exam-Style: Proof with Vectors

Question (A-Level): In triangle OAB, OA = a and OB = b. Point M is the midpoint of AB. Prove that OM = ½(a + b).

Solution: 1. AB = OB - OA = b - a. 2. AM = ½AB = ½(b - a). 3. OM = OA + AM = a + ½(b - a) = a + ½b - ½a = ½a + ½b = ½(a + b). Conclusion: OM = ½(a + b), as required.

What we did and why: - We expressed AM as half of AB, then used vector addition to find OM. - This proves M is the midpoint by showing OM is the average of a and b.

Common Mistakes

1. MISTAKE: Forgetting the square root when finding magnitude.
   WHY IT HAPPENS: Students stop at x² + y² and forget the final step.
   CORRECT APPROACH: Always take the square root of x² + y².

2. MISTAKE: Adding vectors incorrectly (e.g., (2, 3) + (1, 4) = (3, 7) instead of (3, 7)).
   WHY IT HAPPENS: Confusing vector addition with multiplication.
   CORRECT APPROACH: Add x and y components separately.

3. MISTAKE: Mixing up AB and BA (e.g., writing AB = A - B).
   WHY IT HAPPENS: Not remembering that AB = B - A.
   CORRECT APPROACH: Always subtract the start point from the end point.

4. MISTAKE: Assuming vectors are parallel if they look similar (e.g., (2, 4) and (4, 8) are parallel, but (2, 4) and (4, 6) are not).
   WHY IT HAPPENS: Not checking for scalar multiples.
   CORRECT APPROACH: Check if one vector is a multiple of the other (e.g., (4, 8) = 2 × (2, 4)).

5. MISTAKE: Incorrectly proving geometric properties (e.g., saying two vectors are equal when they’re not).
   WHY IT HAPPENS: Skipping steps or misapplying vector addition.
   CORRECT APPROACH: Write every step clearly and check for scalar multiples.

Exam Traps

1. Trap: Giving vectors in column form (e.g., a = (3 4)) but expecting answers in row form (e.g., (3, 4)).
   How to Spot it: The question uses brackets differently.
   How to Avoid it: Stick to the format given in the question.

2. Trap: Asking for a vector in terms of i and j (e.g., 3i + 4j) but the student answers in component form (e.g., (3, 4)).
   How to Spot it: The question specifies i and j.
   How to Avoid it: Convert to i and j if required (e.g., (3, 4) = 3i + 4j).

3. Trap: Hiding a vector proof in a geometry question (e.g., "Prove that ABCD is a parallelogram").
   How to Spot it: The question mentions coordinates or vectors.
   How to Avoid it: Assign vectors to points and check for parallelism or equal lengths.

1-Minute Recap

"Right, listen up—this is your last-minute vector cheat sheet!

1. Magnitude? Square the components, add them, take the square root. Done.
2. Adding vectors? Add x’s, add y’s. That’s it.
3. Vector between points? End minus start. Always.
4. Proving stuff? Assign vectors to points, express everything in terms of those vectors, then compare.
5. Parallel vectors? One must be a multiple of the other. Check the ratio!
6. Exam traps? Watch for i/j notation, column vs. row vectors, and hidden proofs.

Now go smash that exam—you’ve got this!