Vectors: Notation, Addition and Scalar Multiples
Write vectors as columns or as bold letters, add and subtract them along a route, and multiply by a scalar to change length or reverse direction.
What a learner can do afterwards
- Add two column vectors and draw the result on a grid
- Write a route around a shape as a sum of named vectors
- Say what -2a does to the length and the direction of a
- Subtract matching vector components to find a difference
- Add three or more vectors component by component
- Reverse a vector by multiplying every component by -1
- Use a scalar to multiply every component of a vector
- Explain why changing the order of vector addition does not change the sum
1 · Read
A column vector says how far right and how far up to go. Add matching entries: top with top and bottom with bottom. For example, (7, -2) plus (3, 5) is (10, 3).
Add another pair the same way. The top entries in (6, 2) and (4, 7) give 6 plus 4 = 10. The bottom entries give 2 plus 7 = 9, so the sum is (10, 9).
Routes add tip to tail. If vector a takes you from A to B and vector b takes you from B to C, the whole route from A to C is a plus b.
Subtract matching entries, or add the reverse of the second vector. For (9, 6) minus (4, 2), subtract tops and bottoms to get (5, 4).
A scalar multiplies every component. For -3a, the 3 triples the length and the minus reverses direction. If a has length 4, then -3a has length 12.
A positive scalar keeps direction and changes length by its size. If vector b has length 7, multiplying by 3 gives a vector with length 21 and the same direction.
Check a vector sum by adding matching entries again. If the top entries total 6 and the bottom entries total 6, the result must have both components equal to 6. A different component shows an arithmetic error.
Add matching components, chain routes tip to tail, subtract by reversing the second vector, and use scalars to stretch or reverse arrows.
The lesson and quiz items were written for this dataset by its content-upgrade pipeline. The lesson examples and quiz item q8 were later revised.
The rewritten extra items retain their per-item original source records. The rewritten readings were written for this dataset. Unchanged reading and media records retain their listed source credits.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.