Lesson 2 of 8
Add and subtract vectors
Combine two vectors by adding or subtracting their matching components.
The idea
To add two vectors, add their matching components: first with first, second with second. The answer is another vector with the same number of components.
Components only ever combine with the component in the same position. The first component of the answer comes only from the two first components.
What it looks like
Draw u from the origin, then start v where u ends. The arrow from the origin to where v finishes is u + v. This is sometimes called “tip to tail”.
Subtracting
Subtraction works the same way: subtract matching components. Order matters here. In u − v you always start with u and take away v.
Find uminusv.
Pair matching components
u minus v equals the vector with components 5 minus 2, and 4 minus 1Subtract each pair
u minus v equals the vector with components 3, 3
Zeros and negative numbers
Adding 0 changes nothing, so a zero component simply copies the other number. Negative components follow the usual rules of arithmetic. Adding a negative number is the same as subtracting:
Practice starts with positive numbers, then adds zeros, and only then brings in negatives.
Worked example
Find uplusv.
Pair matching components
u plus v equals the vector with components 4 plus 3, and 2 plus 6Add each pair
u plus v equals the vector with components 7, 8
Common mistakes
- Mixing positions. Adding the first component of u to the second component of v. Keep each position separate.
- Subtracting in the wrong order. u − v and v − u give different answers. Start with u.
- Losing a minus sign. In 2 − (−3), subtracting a negative adds: the answer is 5.