The idea

Solving geometric problems with vectors

Using vectors to prove geometry: parallel vectors as scalar multiples, collinear points as parallel vectors through a shared point, dividing a line in a ratio, expressing figures in terms of base vectors, and comparing areas.

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Pure · Vectors

Solving geometric problems with vectors

Why it works

Geometry becomes algebra

Vectors turn geometry into algebra. Three ideas do most of the work.

Parallel. Two non-zero vectors are parallel exactly when one is a scalar multiple of the other:

\vec{AB} = k\,\vec{CD}

. The number

k

also tells you the length ratio —

\vec{AB}

|k|

times as long as

\vec{CD}

Collinear needs a shared point

Three points

A

B

C

lie on one straight line exactly when

\vec{AB}

and

\vec{AC}

are parallel — i.e.

\vec{AB} = k\,\vec{AC}

. Parallel alone is not enough: the two vectors must also share a point (here

A

), or you have proved nothing about whether the lines are the same line. This is the workhorse for "show that

A

B

C

are collinear" and "find the value of

p

for which they lie on a straight line":

\vec{AB} = k\,\vec{AC} \;\Longrightarrow\; A, B, C \text{ collinear}

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