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

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 \[\vec{a} \text{ and } \vec{b}\] .

Find λ, if \[\left( 2 \hat{ i }  + 6 \hat{ j }  + 14 \hat{ k }  \right) \times \left( \hat{ i }  - \lambda \hat{ j } + 7 \hat{ k }  \right) = \vec{0} .\]

Write the value of the area of the parallelogram determined by the vectors   \[2 \hat{ i }  \text{ and } 3 \hat{ j }  .\]

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

Find a vector of magnitude \[\sqrt{171}\]  which is perpendicular to both of the vectors \[\vec{a} = \hat{ i } + 2 \hat{ j }  - 3 \hat{ k } \]  and  \[\vec{a} = \hat{ i } + 2 \hat{ j }  - 3 \hat{ k } \] . 

Write the number of vectors of unit length perpendicular to both the vectors \[\vec{a} = 2 \hat{ i } + \hat{ j }  + 2 \hat{ k }  \text{ and }  \vec{b} = \hat{ j }  + \hat{ k } \] .

Write the angle between the vectors  \[\vec{a} \times \vec{b}\]  and  \[\vec{b} \times \vec{a}\] .

If \[\vec{a}\] is any vector, then \[\left( \vec{a} \times \hat{ i }  \right)^2 + \left( \vec{a} \times \hat{ j } \right)^2 + \left( \vec{a} \times \hat{ k }  \right)^2 =\]

If \[\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}\] and \[\vec{a} \times \vec{b} = \vec{a} \times \vec{c,} \vec{a} \neq 0,\] then

The vector \[\vec{b} = 3 \hat { i }+ 4 \hat {k }\] is to be written as the sum of a vector \[\vec{\alpha}\] parallel to \[\vec{a} = \hat {i} + \hat {j}\] and a vector \[\vec{\beta}\] perpendicular to \[\vec{a}\]. Then \[\vec{\alpha} =\]

The unit vector perpendicular to the plane passing through points \[P\left( \hat{ i } - \hat{ j }  + 2 \hat{ k }  \right), Q\left( 2 \hat{ i } - \hat{ k } \right) \text{ and }  R\left( 2 \hat{ j }  + \hat{ k }  \right)\]  is 

If \[\vec{a,} \vec{b}\] represent the diagonals of a rhombus, then

Vectors \[\vec{a} \text{ and }  \vec{b}\] are inclined at angle θ = 120°. If \[\left| \vec{a} \right| = 1, \left| \vec{b} \right| = 2,\] then  \[\left[ \left( \vec{a} + 3 \vec{b} \right) \times \left( 3 \vec{a} - \vec{b} \right) \right]^2\]  is equal to 

If \[\vec{a} = \hat{ i }  + \hat{ j }  - \hat{ k }  , \vec{b} = - \hat{ i }  + 2\hat{ j }  + 2 \hat{ k }  \text{ and }  \vec{c} = - \hat{ i } + 2 \hat{ j }  - \hat{ k }  ,\]  then a unit vector normal to the vectors \[\vec{a} + \vec{b} \text{ and }  \vec{b} - \vec{c}\]  is

A unit vector perpendicular to both \[\hat{ i }  + \hat{ j } \text{ and }  \hat{ j } + \hat{ k } \] is

If \[\vec{a} = 2 \hat{ i }  - 3 \hat{ j }  - \hat{ k }  \text{ and }  \vec{b} = \hat{ i } + 4 \hat{ j }  - 2 \hat{ k 
} , \text{ then } \vec{a} \times \vec{b}\]  is

If \[\hat{ i }  , \hat{ j }  , \hat{ k } \] are unit vectors, then

If θ is the angle between the vectors \[2 \hat{ i }  - 2 \hat{ j}  + 4 \hat{ k }  \text{ and } 3 \hat{ i }  + \hat { j }  + 2 \hat{ k }  ,\]  then sin θ =

If \[\left| \vec{a} \times \vec{b} \right| = 4, \left| \vec{a} \cdot \vec{b} \right| = 2, \text{ then }  \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 =\]