Vectors can be used to solve problems in geometry. In two dimensions, it is possible to describe the position of any point using two vectors. For example, using the vectors a and b shown in the diagram:
AC = 2a
EF = a
IK = 2a
AF = a + b
AL = 3a + 2b
LE = −3a − b
Note that AB, EF and IK are all parallel vectors as they have the same direction.
Worked Examples
In the parallelogram shown below, AB = b and AD = d. Also X is the midpoint of BC and Y lies on DC such that DY = 2CY.
Express the following vectors in terms of b and d.
AC
Show me| AC | = AD + DC |
| = b + d |
BX
Show me| BX | = BC |
| = d |
AX
Show me| AX | = AB + BX |
| = b + d |
DY
Show me| DY | = DC |
| = b |
AY
Show me| AY | = AD + DY |
| = d + b |
XY
Show me| XY | = XA + AY |
| = −⎛ ⎝b + d⎞ ⎠ + ⎛ ⎝d + b⎞ ⎠ | |
| = b − b + d − d | |
| = −b + d |
The diagram shows 2 identical parallelograms. The vector q = AF and the vector p = AB. The point M is the midpoint of CD.
Show that BM is parallel to AD.
Show me| BM | = BC + CM |
| = p + q |
| AD | = AC + CD |
| = 2p + q |
As AD = 2BM, the lines AD and BM must be parallel.
Show that EM is parallel to FC.
Show me| EM | = ED + DM |
| = p + ⎛ ⎝−q⎞ ⎠ | |
| = p − q |
| FC | = FD + DC |
| = 2p − q |
As FC = 2EM, the lines FC and EM must be parallel.
In the figure above, not drawn to scale, ABCD is a parallelogram such that DC = 3x and DA = 3y. The point P is on DB such that DP : PB = 1:2.
Express in terms of x and y.
Show that AP = x − 2y.
Show me AP = AD + DP = −3y + (x + y) = x − 2yGiven that E is the mid-point of DC, prove that A, P and E are collinear.
Show meDE = x, so PE = PD + DE = −x − y + x = x − y
Hence
2 × PE = x − 2y = AP
that is PE and AP have the same direction. So A, P and E are collinear.
Given that x = and y = , use a vector method to prove that triangle AED is isosceles.
Show meWe need to find two sides of equal length.
Now
DE = x = =
and
DE = = = = 3
AE = AD + DE = −3y + x = −3 +
= +
=
So
AE = √02 + (−3)2 = = = 3
As AE = DE = 3, triangle ADE is isosceles.
Exercises
