हिंदी

If → a , → B Represent the Diagonals of a Rhombus, Then - Mathematics

Advertisements
Advertisements

प्रश्न

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

विकल्प

  • \[\vec{a} \times \vec{b} = \vec{0}\]

  • \[\vec{a} \cdot \vec{b} = 0\]

  • \[\vec{a} \cdot \vec{b} = 1\]

  • \[\vec{a} \times \vec{b} = \vec{a}\]

MCQ

उत्तर

\[\text{ We know that the diagonals in a rhombus }  ( \vec{a} \text{ and }  \vec{b} ) \text{ are perpendicular } .\]

\[\text{ Therefore, their dot product is zero} .\]

\[ \Rightarrow \vec{a} . \vec{b} = 0\]

shaalaa.com
  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
अध्याय 25: Vector or Cross Product - MCQ [पृष्ठ ३५]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 12
अध्याय 25 Vector or Cross Product
MCQ | Q 5 | पृष्ठ ३५

वीडियो ट्यूटोरियलVIEW ALL [2]

संबंधित प्रश्न

Find `|veca × vecb|`, if `veca = hati - 7hatj + 7hatk` and `vecb = 3hati - 2hatj + 2hatk`.


Find λ and μ if  `(2hati + 6hatj + 27hatk) xx (hati + lambdahatj + muhatk) = vec0`.


Find the area of the parallelogram whose adjacent sides are determined by the vector `veca = hati - hatj + 3hatk` and `vecb = 2hati - 7hatj + hatk`.


If A, B, C are three non- collinear points with position vectors `vec a, vec b, vec c`, respectively, then show that the length of the perpendicular from Con AB is `|(vec a xx vec b)+(vec b xx vec c) + (vec b xx  vec a)|/|(vec b -  vec a)|`


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

 


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

 


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.

 
 

\[\text{ If }  \left| \vec{a} \right| = 13, \left| \vec{b} \right| = 5 \text{ and }  \vec{a} . \vec{b} = 60, \text{ then find }  \left| \vec{a} \times \vec{b} \right| .\]

 


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

 

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

 
 

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

 
 

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


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.

 

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 \[- \hat { i }  + 3 \hat{ j }  + 4 \hat{ k } \] .

 

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 \[\vec{a} \text{ and }  \vec{b} .\]

 
 
 

 


\[\text{ If }  \left| \vec{a} \right| = 10, \left| \vec{b} \right| = 2 \text{ and }  \left| \vec{a} \times \vec{b} \right| = 16, \text{ find }  \vec{a} . \vec{b} .\]

 


For any two vectors \[\vec{a}\] and \[\vec{b}\] , find \[\vec{a} . \left( \vec{b} \times \vec{a} \right) .\]

 
 
 
 

For any three vectors \[\vec{a,} \vec{b} \text{ and }  \vec{c}\] write the value of \[\vec{a} \times \left( \vec{b} + \vec{c} \right) + \vec{b} \times \left( \vec{c} + \vec{a} \right) + \vec{c} \times \left( \vec{a} + \vec{b} \right) .\]

 
 

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

 

 


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

 

 


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 

 

The value of \[\left( \vec{a} \times \vec{b} \right)^2\] is 

 

The value of λ for which the two vectors `2hati - hatj + 2hatk` and `3hati + λhatj + hatk` are perpendicular is ______.


Find the area of the triangle with vertices A(1, l, 2), (2, 3, 5) and (1, 5, 5).


Let `veca = hati + hatj, vecb = hati - hatj` and `vecc = hati + hatj + hatk`. If `hatn` is a unit vector such that `veca.hatn` = 0 and `vecb.hatn` = 0, then find `|vecc.hatn|`.


If the vector `vecb = 3hatj + 4hatk` is written as the sum of a vector `vec(b_1)`, parallel to `veca = hati + hatj` and a vector `vec(b_2)`, perpendicular to `veca`, then `vec(b_1) xx vec(b_2)` is equal to ______.


Find the area of a parallelogram whose adjacent sides are determined by the vectors `veca = hati - hatj + 3hatk` and `vecb = 2hati - 7hatj + hatk`.


If `veca xx vecb = veca xx vecc` where `veca, vecb` and `vecc` are non-zero vectors, then prove that either `vecb = vecc` or `veca` and `(vecb - vecc)` are parallel.


If `veca` is a unit vector perpendicular to `vecb` and `(veca + 2vecb).(3veca - vecb) = -5`, find `|vecb|`.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×