Adding vectors, scalar multiples and unit vectors
Add vectors head to tail and with a parallelogram, subtract one vector from another, stretch and flip a vector with a number, and write any vector with i and j.
- UK: Year 10
- France: grade 10
- Iran: grade 8
- Afghanistan: grade 11
- Syria: grade 10
- Palestine: grade 11
- Ukraine: grade 9
- Russia: grade 9
Guide
To add two vectors, put them head to tail: start the second where the first ends. a + b is the arrow from the start of a to the end of b. In numbers, add the tops and add the bottoms: 31 + 13 = 44. If a and b start at the same point, a + b is the diagonal of the parallelogram they make.
a − b means a + (−b): −b is b turned round, the same length the other way. In numbers, subtract the tops and subtract the bottoms.
2a is a stretched to twice the length, the same way: both numbers double. −½a is half as long and points the other way: both numbers are multiplied by −½. The number in front is a scalar.
i = 10 is one step right and j = 01 is one step up: the unit vectors. Every vector is a number of i's plus a number of j's: 3−2 = 3i − 2j. Careful: the length of a + b is usually less than the length of a plus the length of b, because the straight way is shorter than the way round.