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Area of Parallelogram

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AREA OF PARALLELOGRAM
If two sides of a parallelogram are represented by two vectors A and B, then the magnitude of their cross product will be equal to the area of parallelogram i.e.
PROOF
Consider a parallelogram OABC whose two sides are represented by two vectors A and B as shown. The area of parallelogram OABC is equal to
:
= hA
----------------(1)
Draw perpendicular CD on side OA.
Consider right angled triangle COD
sin
= h/OC h
=
OC
sin
h
=
B
sin
Putting the value of h in equation (1), we get,
=
B
sin
A
=
AB
sin
OR
COMMUTATIVE LAW FOR DOT PRODUCT
This law states that :
The scalar product of two vectors and is equal to the magnitude of vector times the projection of
onto the direction of vector .
Consider two vectors and ,the angle between them is
where represents
the projection of vector onto the direction of vector .
Similarly,
Where represents
the projection of vector onto the direction of vector .
Since
This shows that the dot product of two vectors does not chanfe with the change in the order of the vectors to be multiplied. This fact is known as the commutative of dot product.
Consider two vectors and . Let these two vectors represent two adjacent sides of a parallelogram. We construct a parallelogram
OACB
as shown in the diagram. The diagonal
OC
represents the resultant vector From above figure it is clear that:
This fact is referred to as the commutative law of vectr addition
.
ASSOCIATIVE LAW OF VECTOR ADDITION
The law states that the sum of vectors remains same irrespective of their order or grouping in which they are arranged. Consider three vectors , and Applying head to tail rule to obtain the resultant of (+ ) and (+ )
Then finally again find the resultant of these three vectors :
This fact is known as the
ASSOCIATIVE LAW OF VECTOR ADDITION
.

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