Vectors in proof and problem solving
Solve the balance point of weighted points, a geometric proof with vectors (the line joining the midpoints of two sides of a triangle) and problems about speed and parallelograms by drawing vectors.
- UK: Year 11
- Syria: grade 10
- Ukraine: grade 9
- Russia: grade 9
Guide
This extra studio is for students who can add vectors and multiply a vector by a scalar. The barycentre (weighted centre) of points A and B with weights 1 and 2 is the point G with OG = a + 2b3: G is nearer the point with more weight.
In a proof with vectors we write every segment as a vector and simplify with addition and scalar multiples. For example, in a triangle the segment joining the midpoints of two sides is parallel to the third side and half as long.
Common mistakes: taking the midpoint instead of the weighted point, forgetting to halve, and adding lengths instead of components.