Question

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

Answer 1

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 \[\widehat a\]. It is given by \[\widehat a = \dfrac{a}{{\left| a \right|}}\].
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 \[a = i + 2j\] and \[b = 2i + j\]
We know that diagonals of a parallelogram are \[a + b\] and \[a - b\]
So \[a + b = \left( {i + 2j} \right) + \left( {2i + j} \right) = 3i + 3j\]
And \[a - b = \left( {i + 2j} \right) - \left( {2i + j} \right) = - i + j\]
We have to find the angle between the diagonals. We will find this by using the dot product formula which is as follows
\[\left( {a + b} \right).\left( {a - b} \right) = \left( {3i + 3j} \right).\left( { - i + j} \right)\]
Thus we get ,
\[\left( {a + b} \right).\left( {a - b} \right) = 3 - 3 = 0\]
Since the dot product of the diagonals is equal to \[0\] . therefore diagonals are at an angle of \[{90^ \circ }\] 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 \[\widehat a\]. It is given by \[\widehat a = \dfrac{a}{{\left| a \right|}}\].