Science (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

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

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

If \[\vec{a} = 3 \hat{ i }  - \hat{ j }  + 2 \hat{ k } \] and  \[\vec{b} = 2 \hat { i }  + \hat{ j }  - \hat{ k} ,\]  then find \[\left( \vec{a} \times \vec{b} \right) \vec{a} .\]

Write a unit vector perpendicular to \[\hat{ i } + \hat{ j }  \text{ and }  \hat{ j }  + \hat{ k } .\]

If \[\left| \vec{a} \times \vec{b} \right|^2 + \left( \vec{a} . \vec{b} \right)^2 = 144\]  and \[\left| \vec{a} \right| = 4,\]  find \[\left| \vec{b} \right|\] . 

If \[\vec{r} = x \hat{ i } + y \hat{ j }  + z \hat{ k }  ,\] then write the value of \[\left| \vec{r} \times \hat{ i }  \right|^2 .\]

If \[\vec{a} \text{ and }  \vec{b}\] are unit vectors such that \[\vec{a} \times \vec{b}\] is also a unit vector, find the angle between \[\vec{a} \text{ and } \vec{b}\] .

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

If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, then write the value of \[\left| \vec{a} \times \vec{b} \right|^2 + \left( \vec{a} . \vec{b} \right)^2 .\]

If \[\vec{a}\] is a unit vector such that \[\vec{a} \times \hat{ i }  = \hat{ j }  , \text{ find }  \vec{a} . \hat{ i } \] .

If  \[\vec{c}\] is a unit vector perpendicular to the vectors \[\vec{a} \text{ and } \vec{b} ,\]  write another unit vector perpendicular to \[\vec{a} \text{ and }  \vec{b} .\]

Find the angle between two vectors \[\vec{a} \text{ and }  \vec{b}\] with magnitudes 1 and 2 respectively and when \[\left| \vec{a} \times \vec{b} \right| = \sqrt{3} .\]

Vectors  \[\vec{a} \text{ and }  \vec{b}\] \[\left| \vec{a} \right| = \sqrt{3}, \left| \vec{b} \right| = \frac{2}{3}\text{ and } \left( \vec{a} \times \vec{b} \right)\]  is a unit vector. Write the angle between \[\vec{a} \text{ and } \vec{b}\] .