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One side of the parallelogram has length 3 and another side has length 4. Let a and b denote the lengths of the diagonals of the parallelogram. Which of the following quantities can be determined from the given information?
(i) a+b
(ii) a2+b2 
(iii) a3+b3
A) Only (i)
B) Only (ii)
C) Only (iii)
D) Only (i) and (ii)

Answer
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Hint: Two vectors a and brepresented by the two adjacent sides of a parallelogram in magnitude and direction, then their sum a+b is represented in magnitude and direction by the diagonal of the parallelogram through their common path. This is known as the parallelogram law of vector addition.
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The dot product or scalar product of two vectors a and b is the multiplication of the magnitude of two vectors and cos of the angle between them.
ab=|a||b|cosθ,
aa=|a|2 [cos0=1] …………………………...…… (1)

Complete step by step answer:
In parallelogram ABCD, AB || CD, AD || BC …… (from figure 2)
and vectors of parallelogram.
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AB=DC,AD=BC …… (from figure 3)
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AC =a, BD =b, ( given) …… (2)
AB + AD = AC ( parallelogram law of vector addition)
Taking dot product on both sides
(AB + AD)(AB + AD) = ACAC
|AC|2=|AB|2+|AD|2+2(AB.AD) (from (1)) …… (3)
AB  AD = BD ( parallelogram law of vector addition)
Taking dot product on both sides
(ABAD)(ABAD) = BDBD
|BD|2=|AB|2+|AD|22(AB.AD) …… (4)
Adding equations (2) and (3)
|AC|2+|BD|2=2(|AB|2+|AD|2) …… (5)
Magnitudes: |AC|=a; |BD|=b; |AB|=3; |AD|=4; ( given)
a2 +  b2 = 2(32 + 42)
=2(9+16)
=2(25)
=50

Option (B): only (ii) is correct. One can only find the value of a2 + 2 = 50.

Note:
One can remember the result derived in the above solution: The sum of squares of length of diagonals of a parallelogram is equal to the sum of squares of length of all the sides of the parallelogram. The result can help solve such problems.