Representing vectors by their “components” in the coordinate plane makes calculations easier.
Basic Vectors and Components
Using the unit vector →e₁=(1, 0) in the x-axis direction and the unit vector →e₂=(0, 1) in the y-axis direction, any plane vector can be represented uniquely as
→a = a₁→e₁ + a₂→e₂ = (a₁, a₂).
Addition, Subtraction, and Scalar Multiplication by Components
For →a = (a₁, a₂), →b = (b₁, b₂), and a real number k,
→a + →b = (a₁ + b₁, a₂ + b₂)
→a − →b = (a₁ − b₁, a₂ − b₂)
k→a = (ka₁, ka₂)
Example ①
When →a=(3, −2) and →b=(−1, 4), find the components of 2→a − 3→b.
Solution: 2→a=(6, −4), 3→b=(−3, 12). Their difference is (9, −16).
When →a=(3, −2) and →b=(−1, 4), find the components of 2→a − 3→b.
Solution: 2→a=(6, −4), 3→b=(−3, 12). Their difference is (9, −16).
Magnitude and the Vector Between Two Points
|→a| = √(a₁² + a₂²). For A(a₁, a₂) and B(b₁, b₂),
→AB = (b₁ − a₁, b₂ − a₂) （terminal point − initial point）
|→AB| = √((b₁−a₁)² + (b₂−a₂)²)
Example ②
When A(1, −3) and B(4, 1), find the components of →AB and |→AB|.
Solution: →AB=(3, 4), |→AB|=√(9+16)=√25=5.
When A(1, −3) and B(4, 1), find the components of →AB and |→AB|.
Solution: →AB=(3, 4), |→AB|=√(9+16)=√25=5.
Condition for Parallel Vectors (Components)
For →a=(a₁, a₂) ≠ →0 and →b=(b₁, b₂) ≠ →0, →a ∥ →b ⇔ a₁b₂ − a₂b₁ = 0.
Key Points
- Calculate sums, differences, and scalar multiples component by component
- |→a| = √(a₁² + a₂²)
- The components of →AB are（terminal point）−（initial point）
- Parallel condition: a₁b₂ − a₂b₁ = 0
Practice Problems
- When →a=(2,−1) and →b=(−3,4), find the components and magnitude of 3→a + 2→b.
- When A(−2, 5) and B(3, −7), find |→AB|.
- Find k such that →a=(3, k) and →b=(−6, 4) are parallel.
Answers and Explanations
- Answer: (0, 5), magnitude 5
Explanation: 3→a=(6,−3), 2→b=(−6,8). Their sum is (0,5), with magnitude 5. - Answer: 13
Explanation: →AB=(5, −12), magnitude √(25+144)=√169=13. - Answer: k=−2
Explanation: 3×4 − k×(−6)=0 → 12+6k=0 → k=−2.