A vector has both magnitude (size) and direction. In two dimensions we write it as a column vector, with the top number giving movement in the x-direction and the bottom number movement in the y-direction.
Adding, subtracting and scaling
To add or subtract vectors, combine the corresponding components. To multiply by a scalar, multiply every component by that number.
Key idea
If a = (x₁, y₁) and b = (x₂, y₂), then a + b = (x₁+x₂, y₁+y₂) and ka = (kx₁, ky₁).
Magnitude
The magnitude (or modulus) of a vector is its length, found using Pythagoras' theorem. The direction of the arrow does not matter for the length, so a vector and its negative have the same magnitude.
Key idea
For a = (x, y), the magnitude is |a| = √(x2 + y2).
Position and parallel vectors
The position vector of a point gives its displacement from the origin, so the point (4, 3) has position vector (4, 3). Two vectors are parallel when one is a scalar multiple of the other, such as (2, 3) and (4, 6) = 2(2, 3); parallel vectors point along the same line even if their lengths differ.
Worked example
Given a = (3, 4) and b = (1, 2):
a + b = (3+1, 4+2) = (4, 6).
2a = (2×3, 2×4) = (6, 8).
|a| = √(32 + 42) = √(9 + 16) = √25 = 5.
Remember
- Add/subtract component by component.
- Magnitude uses Pythagoras: √(x2 + y2).
- A vector always has both magnitude and direction.