Example – 1
From any two vectors  and  , prove that
When does the equality hold in these cases?
Solution: Consider this figure:
The first two relations follow from the fact that in any triangle, the sum of two sides is greater than the third side:
In ΔABC: AC £ AB + BC (we’ll soon talk about how and when the equality comes)
Þ + £ +
In ΔABC ': AC’ £ AB + BC’ = AB + BC
Þ | - | £ || + ||
In the first relation, the equality can hold only if the two vectors have the same direction; this should be intuitively obvious:
The equality in the second relation holds if the two vectors are exactly opposite:
To prove the third relation, we use in ΔABC in Fig - 11, the geometrical fact that the difference of any two sides of a triangle is less than its third side:
|AB - BC| £ AC
Þ|| | -| || £| + |
The equality holds when and are precisely in the opposite direction
The main point to understand from this example is how easily vector relations follows from corresponding geometrical facts.
Example – 2
Suppose that the vectors and represent two adjacent sides of a regular hexagon. Find the vectors representing the other sides.
Solution: Let the hexagon be A1A2A>sub>3A4A5A6, as shown:
First of all, we note an important geometrical property of a regular hexagon:
Diagonal = 2 × side
Þ A1A4 = 2 × A2A3
Also, since A1A4 || A2 A3, we have
Now we use the triangle law to determine the various sides:
= 2 - ( + )
= -
 = . (only the sense differs; support is parallel to
the support of )
Thus, all sides are expressible in terms of and
Example – 3
What can be interpreted about and if they satisfy the relation:
| + | = | - |
Solution: Make and co-initial so that they form the adjacent sides of a parallelogram:
We have,
| + | = | | = OC
and | - = | |= BA
Thus, the stated relation implies that the two diagonals of the parallelogram OACB are equal, which can only happen if OACB is a rectangle. This implies that and
form the adjacent sides of a rectangle. In other words, and are
perpendicular to each other.
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