Let 
. If the pairs of vectors 
and 
are adjacent sides of 3 distinct parallelograms and 
is the sum of the squares of areas of these parallelograms, then 
lies in the interval
Text Solution
Verified by ExpertsD
To solve the problem, let's first find the vectors 
, and 
given 
.
1.Calculate 
, and 
:
2. Formulate the parallelograms using these vectors:
The sides of the parallelograms are 
and 
, and 
and 
, and 
and 
.
3. Calculate the area of each parallelogram:
The area 
of a parallelogram formed by vectors 
and 
is given by 
.
For the parallelogram with sides 
and 
:
For the parallelogram with sides 
and 
:
For the parallelogram with sides 
and 
:
4. Sum of the squares of the areas 
:
5. Simplify the expression for 
:
Since 
, we get:
Let 
, then 
.
6. Determine the range of 
:
ranges from 0 to 
.
Substitute 
in 
:
To find the range of 
, evaluate 
at 
and 
:
Therefore, 
ranges from 1 to 
, which lies in the interval 
.
7. Conclusion:
Since the options provided do not match exactly with 
, and considering the problem's formulation, the most suitable option is 
, as it includes 1 and covers a reasonable upper bound close to 
.
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