Component form
In a Cartesian plane, i is the unit vector along the positive x-axis and j along the positive y-axis. Any vector can be written r = xi + yj, or as a column r = (x, y). To add or subtract, combine components: (x₁i + y₁j) + (x₂i + y₂j) = (x₁ + x₂)i + (y₁ + y₂)j. A scalar multiple k(xi + yj) = kxi + kyj.
Magnitude and unit vector
The magnitude of r = xi + yj is |r| = √(x² + y²), from Pythagoras. A unit vector in the direction of r is r̂ = r / |r|, which always has magnitude 1. For two points A(x₁, y₁) and B(x₂, y₂), AB = (x₂ − x₁)i + (y₂ − y₁)j.
Key formula
Magnitude: |x i + y j| = √(x² + y²). Unit vector: r̂ = r / |r|. Displacement: AB = (x₂ − x₁) i + (y₂ − y₁) j.
Worked example
Given r = 3i + 4j, find |r| and the unit vector r̂. |r| = √(3² + 4²) = √(9 + 16) = √25 = 5. So r̂ = (3i + 4j)/5 = 0.6i + 0.8j. Check: √(0.6² + 0.8²) = √(0.36 + 0.64) = √1 = 1.
Testing for parallel vectors
In component form, two vectors are parallel when one is a scalar multiple of the other. So x₁i + y₁j is parallel to x₂i + y₂j when x₁/x₂ = y₁/y₂, equivalently x₁y₂ − x₂y₁ = 0. For example, 2i + 3j and 4i + 6j are parallel because 4i + 6j = 2(2i + 3j). To find a vector of a given length in a certain direction, first find the unit vector, then multiply by the required length. A vector of length 15 in the direction of 3i + 4j is 15 × (3i + 4j)/5 = 9i + 12j.
Remember
- Add/subtract vectors component by component.
- |r| = √(x² + y²) is never negative.
- Unit vector = divide by its magnitude.