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

For any two vectors \[\vec{a} \text{ and }  \vec{b}\] , prove that \[\left| \vec{a} \times \vec{b} \right|^2 = \begin{vmatrix}\vec{a} . \vec{a} & & \vec{a} . \vec{b} \\ \vec{b} . \vec{a} & & \vec{b} . \vec{b}\end{vmatrix}\]

Define  \[\vec{a} \times \vec{b}\] and prove that \[\left| \vec{a} \times \vec{b} \right| = \left( \vec{a} . \vec{b} \right)\] tan θ, where θ is the angle between \[\vec{a} \text{ and }  \vec{b}\] .

Find the area of the triangle formed by O, A, B when \[\vec{OA} = \hat{ i } + 2 \hat{ j }  + 3 \hat{ k }  , \vec{OB} = - 3 \hat{ i }  - 2 \hat{ j }+ \hat{ k }  .\]

Let \[\vec{a} = \hat{ i } + 4 \hat{ j }  + 2 \hat{ k } , \vec{b} = 3 \hat{ i }- 2 \hat{ j } + 7 \hat{ k }  \text{ and } \vec{c} = 2 \hat{ i } - \hat{ j }  + 4 \hat{ k }  .\]  Find a vector \[\vec{d}\] which is perpendicular to both \[\vec{a} \text{ and } \vec{d}\] \[\text{ and }  \vec{c} \cdot \vec{d} = 15 .\]

Find a unit vector perpendicular to each of the vectors \[\vec{a} + \vec{b} \text{ and }  \vec{a} - \vec{b} , \text{ where }  \vec{a} = 3 \hat{ i }  + 2 \hat{ j }  + 2 \hat{ k }  \text{ and }  \vec{b} = \hat{ i } + 2 \hat{ j }  - 2 \hat{ k }  .\]

Using vectors find the area of the triangle with vertices, A (2, 3, 5), B (3, 5, 8) and C (2, 7, 8).

If \[\vec{a} = 2 \hat{ i } - 3 \hat{ j  } + \hat{ k } , \vec{b} = -\hat{  i }  + \hat{ k } , \vec{c} = 2 \hat{ j }  - \hat{ k } \]  are three vectors, find the area of the parallelogram having diagonals \[\left( \vec{a} + \vec{b} \right)\]  and \[\left( \vec{b} + \vec{c} \right)\] .

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