Advertisements
Advertisements
प्रश्न
Find a unit vector perpendicular to the plane containing the vectors \[\vec{a} = 2 \hat{ i } + \hat{ j } + \hat{ k } \text{ and } \vec{b} = \hat{ i } + 2 \hat{ j } + \hat{ k } .\]
Advertisements
उत्तर
\[ \text{ Given } : \]
\[ \vec{a} = 2\hat{ i } + \hat{ j } + \hat{ k } \]
\[ \vec{b} = \hat{ i } +2 \hat { j } +\text{ k } \]
\[ \therefore \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & 1 & 1 \\ 1 & 2& 1\end{vmatrix}\]
\[ = \left( 1 - 2 \right) \hat{ i } - \left( 2 - 1 \right) \hat{ j } + \left( 4 - 1 \right) \hat{ k } \]
\[ = - \hat{ i } - \hat{ j } + 3 \hat{ k } \]
\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = \sqrt{1 + 1 + 9 }\]
\[ = \sqrt{11}\]
\[\text{ Unit vector perpendicular to the plane containing vectors } \vec{a} \text{ and } \vec{b} = \pm \frac{\vec{a} \times \vec{b}}{\left| \vec{a} \times \vec{b} \right|}\]
\[\text{ Unit vector perpendicular to the plane containing vectors } \vec{a} \text{ and } \vec{b} = \pm \frac{1}{\sqrt{11}}\left( - \hat{ i } - \hat{ j } + 3 \hat{ k } \right)\]
APPEARS IN
संबंधित प्रश्न
If `veca = 2hati + 2hatj + 3hatk, vecb = -veci + 2hatj + hatk and vecc = 3hati + hatj` are such that `veca + lambdavecb` is perpendicular to `vecc`, then find the value of λ.
Find a unit vector perpendicular to each of the vector `veca + vecb` and `veca - vecb`, where `veca = 3hati + 2hatj + 2hatk` and `vecb = hati + 2hatj - 2hatk`.
Find λ and μ if `(2hati + 6hatj + 27hatk) xx (hati + lambdahatj + muhatk) = vec0`.
Given that `veca.vecb = 0` and `veca xx vecb = 0` What can you conclude about the vectors `veca and vecb`?
Let the vectors `veca, vecb, vecc` given as `a_1hati + a_2hatj + a_3hatk, b_1hati + b_2hatj + b_3hatk, c_1hati + c_2hatj + c_3hatk` Then show that = `veca xx (vecb+ vecc) = veca xx vecb + veca xx vecc.`
Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).
Find a unit vector perpendicular to both the vectors \[\vec{a} + \vec{b} \text { and } \vec{a} - \vec{b}\] ,where \[\vec{a} = \hat{i}+ \hat{j} + \hat{k} , \vec{b} =\hat {i} + 2 \hat{j} + 3 \hat{k}\].
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 \[\left| \vec{a} \times \vec{b} \right| = \vec{a} \cdot \vec{b} .\]
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 \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} .\]
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 \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} , \text{ where } \vec{a} = 3 \hat{ i } + 2 \hat{ j } + 2 \hat{ k } \text{ and } \vec{b} = \hat{ i } + 2 \hat{ j } - 2 \hat{ k } .\]
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.
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 value \[\left( \hat{ i } \times \hat{ j } \right) \cdot \hat{ k } + \hat{ i } \cdot \hat{ j } .\]
Write the value of \[\hat{ i } . \left( \hat{ j } \times \hat{ k } \right) + \hat{ j } . \left( \hat{ k } \times \hat{ i } \right) + \hat{ k } . \left( \hat{ i } \times \hat{ j } \right) .\]
Write the value of \[\hat{ i } × \left( \hat{ j } + \hat{ k } \right) + \hat{ j } × \left( \hat{ k } + \hat{ i } \right) + \hat{ k } × \left( \hat{ i } + \hat{ j } \right) .\]
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 such that \[\left| \vec{a} \times \vec{b} \right| = \sqrt{3}\text{ and } \vec{a} . \vec{b} = 1,\] find the angle between.
For any two vectors \[\vec{a} \text{ and } \vec{b} , \text{ find } \left( \vec{a} \times \vec{b} \right) . \vec{b} .\]
Write a unit vector perpendicular to \[\hat{ i } + \hat{ j } \text{ and } \hat{ j } + \hat{ k } .\]
If \[\vec{r} = x \hat{ i } + y \hat{ j } + z \hat{ k } ,\] then write the value of \[\left| \vec{r} \times \hat{ i } \right|^2 .\]
Write the number of vectors of unit length perpendicular to both the vectors \[\vec{a} = 2 \hat{ i } + \hat{ j } + 2 \hat{ k } \text{ and } \vec{b} = \hat{ j } + \hat{ k } \] .
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
If \[\hat{ i } , \hat{ j } , \hat{ k } \] are unit vectors, then
If \[\left| \vec{a} \times \vec{b} \right| = 4, \left| \vec{a} \cdot \vec{b} \right| = 2, \text{ then } \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 =\]
The value of \[\hat{ i } \cdot \left( \hat{ j } \times \hat{ k } \right) + \hat{ j } \cdot \left( \hat{ i } \times \hat{ k } \right) + \hat{ k } \cdot \left( \hat{ i } \times \hat{ j } \right),\] is
If θ is the angle between any two vectors `bara` and `barb` and `|bara · barb| = |bara xx barb|` then θ is equal to ______.
The value of λ for which the two vectors `2hati - hatj + 2hatk` and `3hati + λhatj + hatk` are perpendicular is ______.
Let `veca, vecb, vecc` be three vectors mutually perpendicular to each other and have same magnitude. If a vector `vecr` satisfies. `veca xx {(vecr - vecb) xx veca} + vecb xx {(vecr - vecc) xx vecb} + vecc xx {(vecr - veca) xx vecc} = vec0`, then `vecr` 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`.
Find the area of the parallelogram whose diagonals are `hati - 3hatj + hatk` and `hati + hatj + 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` and `vecb` are two non-zero vectors such that `|veca xx vecb| = veca.vecb`, find the angle between `veca` and `vecb`.
If `veca` is a unit vector perpendicular to `vecb` and `(veca + 2vecb).(3veca - vecb) = -5`, find `|vecb|`.
