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

Find a vector whose length is 3 and which is perpendicular to the vector \[\vec{a} = 3 \hat{ i }  + \hat{ j  } - 4 \hat{ k }  \text{ and }  \vec{b} = 6 \hat{ i }  + 5 \hat{ j }  - 2 \hat{ k } .\]

Find the area of the parallelogram determined by the vector \[2 \hat{ i }  \text{ and }  3 \hat{ j } \] .

Find the area of the parallelogram determined by the vector \[2 \hat{ i } + \hat{ j } + 3 \hat{ k }  \text{ and }  \hat{ i }  - \hat{ j } \] .

Find the area of the parallelogram determined by the vector \[3 \hat{ i } + \hat{ j }  - 2 \hat{ k } \text{  and }  \hat{ i }  - 3 \hat{ j }  + 4 \hat{ k } \] .

Find the area of the parallelogram determined by the vector \[\hat{ i }  - 3 \hat{ j } + \hat{ k }  \text{ and }  \hat{ i }  + \hat{ j } + \hat{ k }  .\]

Find the area of the parallelogram whose diagonals are  \[4 \hat{ i } - \hat{ j }  - 3 \hat{ k }  \text{ and }  - 2 \hat{ j }  + \hat{ j }  - 2 \hat{ k } \]

Find the area of the parallelogram whose diagonals are  \[2 \hat{ i }+ \hat{ k } \text{ and } \hat{ i } + \hat{ j } + \hat{ k } \]

Find the area of the parallelogram whose diagonals are  \[3 \hat{ i }  + 4 \hat{ j }  \text{ and } \hat{ i } + \hat{ j } + \hat{ k }\]

Find the area of the parallelogram whose diagonals are \[2 \hat{ i }  + 3 \hat{ j } + 6 \hat{ k } \text{ and }  3 \hat{ i }  - 6 \hat{ j }  + 2 \hat{ k } \]

If \[\vec{a} = 2 \hat{ i }  + 5 \hat{ j }  - 7 \hat{ k }  , \vec{b} = - 3 \hat{ i } + 4 \hat{ j }  + \hat{ k }  \text{ and } \vec{c} = \hat{ i }  - 2 \hat{ j }  - 3 \hat{ k }  ,\] compute \[\left( \vec{a} \times \vec{b} \right) \times \vec{c} \text{ and }  \vec{a} \times \left( \vec{b} \times \vec{c} \right)\]  and verify that these are not equal.

Given \[\vec{a} = \frac{1}{7}\left( 2 \hat{ i } + 3 \hat{ j } + 6 \hat{ k }  \right), \vec{b} = \frac{1}{7}\left( 3 \hat{ i } - 6 \hat{ j }  + 2 \hat{ k }  \right), \vec{c} = \frac{1}{7}\left( 6 \hat{ i } + 2 \hat{ j }  - 3 \hat{ k }\right), \hat{ i } , \hat{ j }  , \hat{ k } \] being a right handed orthogonal system of unit vectors in space, show that \[\vec{a} , \vec{b} , \vec{c}\] is also another system.

Find the angle between two vectors \[\vec{a} \text{ and }  \vec{b}\] , if \[\left| \vec{a} \times \vec{b} \right| = \vec{a} \cdot \vec{b} .\]

if \[\vec{a} \times \vec{b} = \vec{b} \times \vec{c} \neq 0,\]  then  show that \[\vec{a} + \vec{c} = m \vec{b} ,\]  where m is any scalar.

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 \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} .\]