Advertisements
Advertisements
प्रश्न
Write the value of \[\hat{ i } . \left( \hat{ j } \times \hat{ k } \right) + \hat{ j } . \left( \hat{ k } \times \hat{ i } \right) + \hat{ k } . \left( \hat{ i } \times \hat{ j } \right) .\]
उत्तर
\[\hat{ i } . \left( \hat{ j } \times \hat{ k } \right) + \hat{ j } . \left( \hat{ k } \times \hat{ i } \right) + \hat{ k } . \left( \hat{ i } \times \hat{ j } \right)\]
\[ = \hat{ i } . \hat{ i} + \hat{ j} . j + \hat{ k } . \hat{ k } \]
\[ = \left| \hat{ i } \right|^2 + \left| \hat{ j } \right|^2 + \left| \hat{ k } \right|^2 \]
= 1 + 1 - 1
= 3
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.
Let `veca = 4hati + 5hatj - hatk`, `vecb = hati - 4hatj + 5hatk` and `vecc = 3hati + hatj - hatk`. Find a vector `vecd` which is perpendicular to both `vecc` and `vecb and vecd.veca = 21`
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)|`
If \[\vec{a} = 2 \hat{ i } + \hat{ k } , \vec{b} = \hat { i } + \hat{ j } + \hat{ k } ,\] find the magnitude of \[\vec{a} \times \vec{b} .\]
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 \[3 \hat{ i } + \hat{ j } - 2 \hat{ k } \text{ and } \hat{ i } - 3 \hat{ j } + 4 \hat{ k } \] .
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.
What inference can you draw if \[\vec{a} \times \vec{b} = \vec{0} \text{ and } \vec{a} \cdot \vec{b} = 0 .\]
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 \[\vec{a} \text{ and } \vec{b}\] .
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 \[\vec{a} = a_1 \hat{ i } + a_2 \hat{ j } + a_3 \hat{ k } , \vec{b} = b_1 \hat{ i } + b_2 \hat{ j } + b_3 \hat{ k } \text{ and } \vec{c} = c_1 \hat{ i } + c_2 \hat{ j } + c_3 \hat{ k } ,\]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) .
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.
Write a unit vector perpendicular to \[\hat{ i } + \hat{ j } \text{ and } \hat{ j } + \hat{ k } .\]
If \[\left| \vec{a} \times \vec{b} \right|^2 + \left( \vec{a} . \vec{b} \right)^2 = 144\] and \[\left| \vec{a} \right| = 4,\] find \[\left| \vec{b} \right|\] .
If \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left| \vec{a} . \vec{b} \right| = \left| \vec{a} \times \vec{b} \right|,\] write the angle between \[\vec{a} \text{ and } \vec{b} .\]
If \[\vec{c}\] is a unit vector perpendicular to the vectors \[\vec{a} \text{ and } \vec{b} ,\] write another unit vector perpendicular to \[\vec{a} \text{ and } \vec{b} .\]
Find λ, if \[\left( 2 \hat{ i } + 6 \hat{ j } + 14 \hat{ k } \right) \times \left( \hat{ i } - \lambda \hat{ j } + 7 \hat{ k } \right) = \vec{0} .\]
Find a vector of magnitude \[\sqrt{171}\] which is perpendicular to both of the vectors \[\vec{a} = \hat{ i } + 2 \hat{ j } - 3 \hat{ k } \] and \[\vec{a} = \hat{ i } + 2 \hat{ j } - 3 \hat{ k } \] .
Write the angle between the vectors \[\vec{a} \times \vec{b}\] and \[\vec{b} \times \vec{a}\] .
If \[\vec{a,} \vec{b}\] represent the diagonals of a rhombus, 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
(a) If `veca = hati - 2j + 3veck , vecb = 2hati + 3hatj - 5hatk,` prove that `veca and vecaxxvecb` are perpendicular.
The value of λ for which the two vectors `2hati - hatj + 2hatk` and `3hati + λhatj + hatk` are perpendicular is ______.
The number of vectors of unit length perpendicular to the vectors `vec"a" = 2hat"i" + hat"j" + 2hat"k"` and `vec"b" = hat"j" + hat"k"` is ______.
Let `veca` and `vecb` be two unit vectors and θ is the angle between them, Then `veca + vecb` is a unit vector if-
The two adjacent sides of a parallelogram are represented by vectors `2hati - 4hatj + 5hatk` and `hati - 2hatj - 3hatk`. Find the unit vector parallel to one of its diagonals, Also, find the area of the parallelogram.
If `|veca xx vecb| = sqrt(3)` and `veca.vecb` = – 3, then angle between `veca` and `vecb` is ______.
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` is a unit vector perpendicular to `vecb` and `(veca + 2vecb).(3veca - vecb) = -5`, find `|vecb|`.
