Worked example 1
Add vectors
a = (3, −2), b = (5, 4). Find a + b.
- Add x-components.
- Add y-components.
Answer: a + b = (8, 2).
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Higher Mathematics
Vector operations and magnitude
Add, subtract and scale vectors, then calculate magnitudes from components.
Before you start
- Add and subtract signed numbers accurately.
- Use Pythagoras for a component vector.
- Keep vector components in the correct order.
Explanation
Vectors describe movement with direction and size. Components are added, subtracted and scaled component by component.
Magnitude is the length of the vector and is found using Pythagoras.
Visual support
Method and rules
- If a = (p, q), then |a| = √(p² + q²).
- ka = (kp, kq)
- a + b = (a₁ + b₁, a₂ + b₂).
Worked examples
Worked example 2
Find a magnitude
Find the magnitude of v = (6, 8).
- Use |v| = √(6² + 8²).
- Calculate √100
So: |v| = 10
Worked example 3
Use a scalar multiple
Find 3a for a = (-2, 5).
- Multiply both components by 3.
Answer: 3a = (-6, 15)
Watch out
- Adding the first component of one vector to the second component of another.
- Forgetting that magnitude is always non-negative.
- Scaling only one component.
Exam reminder
Use component notation consistently and state magnitudes as positive lengths. Exact surd answers are often better than rounded decimals unless a rounding instruction is given.