Arts (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

The two adjacent sides of a parallelogram are \[2 \hat{ i  } - 4 \hat{ j }  + 5 \hat{ k }  \text{ and }  \hat{ i } - 2 \hat{ j }  - 3\hat{ k }  .\]\  Find the unit vector parallel to one of its diagonals. Also, find its area. 

If either  \[\vec{a} = \vec{0} \text{ or }  \vec{b} = \vec{0} , \text{ then }  \vec{a} \times \vec{b} = \vec{0} .\]  Is the converse true? Justify your answer with an example.

If \[\vec{a} = a_1 \hat{ i } + a_2 \hat{ j } + a_3 \hat{ k }  , \vec{b} = b_1 \hat{ i }  + b_2 \hat{ j }  + b_3 \hat{ k }  \text{ and }  \vec{c} = c_1 \hat{ i } + c_2 \hat{ j }  + c_3 \hat{ k }  ,\]then verify that \[\vec{a} \times \left(  \vec{b} + \vec{c} \right) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c} .\]

Using vectors, find the area of the triangle with vertice A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5) .

Using vectors, find the area of the triangle with vertice A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1)  .    

Find all vectors of magnitude \[10\sqrt{3}\] that are perpendicular to the plane of \[\hat{ i }  + 2 \hat{ j }  + \hat{ k } \] and \[- \hat { i }  + 3 \hat{ j }  + 4 \hat{ k } \] .

If  \[\left| \vec{a} \times \vec{b} \right|^2 + \left| \vec{a} \cdot \vec{b} \right|^2 = 400\] and  \[\left| \vec{a} \right| = 5,\]  then write the value of \[\left| \vec{b} \right| .\]

Write the value  \[\left( \hat{ i }  \times \hat{ j }  \right) \cdot \hat{ k }  + \hat{ i }  \cdot \hat{ j }  .\]

Write the value of  \[\hat{ i } . \left( \hat{ j } \times \hat{ k }  \right) + \hat{ j }  . \left( \hat{ k } \times \hat{ i }  \right) + \hat{ k }  . \left( \hat{ j }  \times \hat{ i }  \right) .\]

Write the value of  \[\hat{ i } . \left( \hat{ j } \times \hat{ k }  \right) + \hat{ j }  . \left( \hat{ k } \times \hat{ i }  \right) + \hat{ k }  . \left( \hat{ i }  \times \hat{ j }  \right) .\]

Write the value of \[\hat{ i }  × \left( \hat{ j } + \hat{ k }  \right) + \hat{ j }   ×  \left( \hat{ k } + \hat{ i }  \right) + \hat{ k }  ×   \left( \hat{ i }  + \hat{ j }  \right) .\]

Write the expression for the area of the parallelogram having \[\vec{a} \text{ and } \vec{b}\] as its diagonals.

For any two vectors  \[\vec{a} \text{ and }  \vec{b}\] write the value of \[\left( \vec{a} . \vec{b} \right)^2 + \left| \vec{a} \times \vec{b} \right|^2\] in terms of their magnitudes.

If \[\vec{a} \text{ and }  \vec{b}\] are two vectors of magnitudes 3 and \[\frac{\sqrt{2}}{3}\]  espectively such that \[\vec{a} \times \vec{b}\] is a unit vector. Write the angle between \[\vec{a} \text{ and }  \vec{b} .\]

For any two vectors \[\vec{a}\] and \[\vec{b}\] , find \[\vec{a} . \left( \vec{b} \times \vec{a} \right) .\]

If   \[\vec{a} \text{ and }  \vec{b}\] are two vectors such that \[\left| \vec{a} \times \vec{b} \right| = \sqrt{3}\text{ and }  \vec{a} . \vec{b} = 1,\]  find the angle between.

For any three vectors \[\vec{a,} \vec{b} \text{ and }  \vec{c}\] write the value of \[\vec{a} \times \left( \vec{b} + \vec{c} \right) + \vec{b} \times \left( \vec{c} + \vec{a} \right) + \vec{c} \times \left( \vec{a} + \vec{b} \right) .\]

For any two vectors \[\vec{a} \text{ and } \vec{b} , \text{ find } \left( \vec{a} \times \vec{b} \right) . \vec{b} .\]