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Question: The adjacent sides of a parallelogram are \[a = i + 2j\] and \[b = 2i + j\] , where \[i\] and \[j\] ...

The adjacent sides of a parallelogram are a=i+2ja = i + 2j and b=2i+jb = 2i + j , where ii and jj are the usual unit vectors along the positive directions of x and y-axes respectively. Then the angle between the diagonals is?
1. 30∘{30^ \circ } and 150∘{150^ \circ }
2. 45∘{45^ \circ } and 135∘{135^ \circ }
3. 60∘{60^ \circ } and 120∘{120^ \circ }
4. 90∘{90^ \circ } and 90∘{90^ \circ }

Explanation

Solution

Hint : A parallelogram is a quadrilateral whose opposite sides are parallel.The opposite angles of a parallelogram are equal.The opposite sides of a parallelogram are equal.The diagonals of a parallelogram bisect each other.

Complete step-by-step answer :
A vector is a quantity that has both magnitude, as well as direction. A vector that has a magnitude of 1 is a unit vector. It is also known as Direction Vector.
Unit Vector is represented by the symbol ^ which is called a cap or hat, such as a^\widehat a. It is given by a^=a∣a∣\widehat a = \dfrac{a}{{\left| a \right|}}.
Unit vectors are usually determined to form the base of a vector space. Every vector in the space can be expressed as a linear combination of unit vectors. The dot product of two unit vectors is a scalar quantity whereas the cross product of two arbitrary unit vectors results in a third vector orthogonal to both of them.
Given the adjacent sides of a parallelogram are a=i+2ja = i + 2j and b=2i+jb = 2i + j
We know that diagonals of a parallelogram are a+ba + b and a−ba - b
So a+b=(i+2j)+(2i+j)=3i+3ja + b = \left( {i + 2j} \right) + \left( {2i + j} \right) = 3i + 3j
And a−b=(i+2j)−(2i+j)=−i+ja - b = \left( {i + 2j} \right) - \left( {2i + j} \right) = - i + j
We have to find the angle between the diagonals. We will find this by using the dot product formula which is as follows
(a+b).(a−b)=(3i+3j).(−i+j)\left( {a + b} \right).\left( {a - b} \right) = \left( {3i + 3j} \right).\left( { - i + j} \right)
Thus we get ,
(a+b).(a−b)=3−3=0\left( {a + b} \right).\left( {a - b} \right) = 3 - 3 = 0
Since the dot product of the diagonals is equal to 00 . therefore diagonals are at an angle of 90∘{90^ \circ } from each other .
Therefore option (4) is the correct answer.
So, the correct answer is “Option 4”.

Note: A vector is a quantity that has both magnitude, as well as direction. A vector that has a magnitude of 1 is a unit vector. It is also known as Direction Vector.
Unit Vector is represented by the symbol ^ which is called a cap or hat, such as a^\widehat a. It is given by a^=a∣a∣\widehat a = \dfrac{a}{{\left| a \right|}}.