Use position vectors relative to one origin. If A,B have position vectors a, b, then AB= b- a. Equal displacement vectors give equal lengths and directions. Midpoint theorem In triangle ABC, let M and N be the midpoints of AB and AC. MN= a+ c2- a+ b2 = 12(c- b) = 12 BC. Thus MN BC and MN= 12BC. To prove collinearity, express two displacement vectors from a common point as scalar multiples; to prove a parallelogram, show a pair of opposite directed sides is equal.

Use position vectors relative to one origin. If \(A,B\) have position vectors \(\mathbf a,\mathbf b\), then \(\overrightarrow{AB}=\mathbf b-\mathbf a\). Equal displacement vectors give equal lengths and directions.

In triangle \(ABC\), let \(M\) and \(N\) be the midpoints of \(AB\) and \(AC\).

\(\begin{aligned}\overrightarrow{MN}&=\dfrac{\mathbf a+\mathbf c}{2}-\dfrac{\mathbf a+\mathbf b}{2}\\&=\dfrac12(\mathbf c-\mathbf b)\\&=\dfrac12\overrightarrow{BC}.\end{aligned}\)

Thus \(MN\parallel BC\) and \(MN=\dfrac12BC\). To prove collinearity, express two displacement vectors from a common point as scalar multiples; to prove a parallelogram, show a pair of opposite directed sides is equal.

Vectors

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Geometrical Proof with Vectors (HL only)

Geometrical Proof with Vectors (HL only)