Writing a vector in terms of two vectors
Any vector in the plane can be built from two vectors that do not lie on one line: p = xa + yb. Draw the parts on the grid, find the coefficients, write AM in a parallelogram and find the length of a projection.
- Russia: grade 9
Guide
Take two vectors a and b that are not parallel. Then every vector p in the plane can be written in exactly one way as p = xa + yb. The numbers x and y are the coefficients. If a and b were parallel you could only reach vectors along that direction.
To find x and y, compare the two coordinates: that gives two equations. In a parallelogram, go from the corner along two sides instead, for example AM = AB + BM. Common slips: taking half a side as the whole side, and losing the sign of a coefficient.
The length of the projection of a on b is (a·b) : |b|: the dot product divided by the length of b. This is a more advanced idea.