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Problem solving with vectors involves using these vector proofs to help us to find out
additional information
Parallel vectors
Two vectors are parallel if and only if one is a multiple of the other
This tends to appear in vector proofs in the following ways:
If you find in your workings that one vector is a multiple of the other, then you know
that the two vectors are parallel – you can then use that fact in the rest of the proof
If you need to show that two vectors are parallel, then all you need to do is show that
one of the vectors multiplied by some number is equal to the other one
Points on a straight line
Often you are asked to show in a vector proof that three points lie on a straight line
(ie that they are collinear)
To show that three points a, b and c lie on a straight line,
show that the vectors connecting the three points are parallel,
for example, show that AB is a multiple of (and therefore parallel to) AC, or that AB is
a multiple of (and therefore parallel) to BC
as those two vectors are parallel and they share a common point it means that the
three lines form a straight line
Vectors divided in ratios
Be careful turning ratios into fractions in vector proofs!
If a point divides a line segment in the ratio p : q, then: