PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

if \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 7 \text{ and }  \vec{a} \times \vec{b} = 3 \hat{ i }  + 2 \hat{ j } + 6 \hat{ k } ,\]  find the angle between  \[\vec{a} \text{ and }  \vec{b} .\]

What inference can you draw if \[\vec{a} \times \vec{b} = \vec{0} \text{ and }  \vec{a} \cdot \vec{b} = 0 .\]

If \[\vec{a,} \vec{b,} \vec{c}\] are three unit vectors such that \[\vec{a} \times \vec{b} = \vec{c} , \vec{b} \times \vec{c} = \vec{a,} \vec{c} \times \vec{a} = \vec{b} .\]  Show that \[\vec{a,} \vec{b,} \vec{c}\] form an orthonormal right handed triad of unit vectors.

Find a unit vector perpendicular to the plane ABC, where the coordinates of A, B and Care A (3, −1, 2), B (1, −1, −3) and C (4, −3, 1).

If a, b, c are the lengths of sides, BC, CA and AB of a triangle ABC, prove that \[\vec{BC} + \vec{CA} + \vec{AB} = \vec{0}\]  and deduce that \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} .\]

if \[\vec{a} = \hat{ i }- 2\hat{ j }  + 3 \hat{ k }  , \text{ and }  \vec{b} = 2 \hat{ i }  + 3 \hat{ j }  - 5 \hat{ k }  ,\]  then find \[\vec{a} \times \vec{b} .\]  Verify th at \[\vec{a} \text{ and }  \vec{a} \times \vec{b}\] are perpendicular to each other.

If  \[\vec{p} \text{ and }  \vec{q}\] are unit vectors forming an angle of 30°; find the area of the parallelogram having \[\vec{a} = \vec{p} + 2 \vec{q} \text{ and }  \vec{b} = 2 \vec{p} + \vec{q}\] as its diagonals.

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)  .