Advertisements
Advertisements
प्रश्न
Find `|veca × vecb|`, if `veca = hati - 7hatj + 7hatk` and `vecb = 3hati - 2hatj + 2hatk`.
Advertisements
उत्तर
`veca = hati - 7hatj + 7hatk, vecb = 3hati - 2hatj + 2hatk`
`veca xx vecb = |(hati, hatj, hatk), (1, -7, 7), (3, -2, 2)|`
= `hati(-14 + 14) - hatj(2 - 21) + hatk(-2 + 21)`
= `19hatj + 19hatk`
∴ `|veca xx vecb| = sqrt((19)^2 + (19)^2)`
`= 19sqrt(1 + 1)`
`= 19sqrt2`.
APPEARS IN
संबंधित प्रश्न
If a unit vector `veca` makes an angles `pi/3` with `hati, pi/4` with `hatj` and an acute angle θ with `hatk`, then find θ and, hence the compounds of `veca`.
Let the vectors `veca` and `vecb` be such that `|veca| = 3` and `|vecb| = sqrt2/3`, then `veca xx vecb` is a unit vector, if the angle between `veca` and `vecb` is ______.
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 \[\vec{a} = 3 \hat { i } + 4 \hat { j } \text{ and } \vec{b} = \hat { i } + \hat{ j } + \hat{ k } ,\] find the value of \[\left| \vec{a} \times \vec{b} \right| .\]
Find a unit vector perpendicular to the plane containing the vectors \[\vec{a} = 2 \hat{ i } + \hat{ j } + \hat{ k } \text{ and } \vec{b} = \hat{ i } + 2 \hat{ j } + \hat{ k } .\]
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 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).
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 \[\vec{a} \text{ and } \vec{d}\] \[\text{ and } \vec{c} \cdot \vec{d} = 15 .\]
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, 2, 3), B(2, −1, 4) and C(4, 5, −1) .
Define vector product of two vectors.
Write the value of \[\hat{ i } . \left( \hat{ j } \times \hat{ k } \right) + \hat{ j } . \left( \hat{ k } \times \hat{ i } \right) + \hat{ k } . \left( \hat{ j } \times \hat{ i } \right) .\]
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 \[\left| \vec{a} . \vec{b} \right| = \left| \vec{a} \times \vec{b} \right|,\] write the angle between \[\vec{a} \text{ and } \vec{b} .\]
If \[\vec{a}\] is a unit vector such that \[\vec{a} \times \hat{ i } = \hat{ j } , \text{ find } \vec{a} . \hat{ i } \] .
Find λ, if \[\left( 2 \hat{ i } + 6 \hat{ j } + 14 \hat{ k } \right) \times \left( \hat{ i } - \lambda \hat{ j } + 7 \hat{ k } \right) = \vec{0} .\]
Write the angle between the vectors \[\vec{a} \times \vec{b}\] and \[\vec{b} \times \vec{a}\] .
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
If \[\vec{a,} \vec{b}\] represent the diagonals of a rhombus, then
Vectors \[\vec{a} \text{ and } \vec{b}\] are inclined at angle θ = 120°. If \[\left| \vec{a} \right| = 1, \left| \vec{b} \right| = 2,\] then \[\left[ \left( \vec{a} + 3 \vec{b} \right) \times \left( 3 \vec{a} - \vec{b} \right) \right]^2\] is equal to
If \[\vec{a} = \hat{ i } + \hat{ j } - \hat{ k } , \vec{b} = - \hat{ i } + 2\hat{ j } + 2 \hat{ k } \text{ and } \vec{c} = - \hat{ i } + 2 \hat{ j } - \hat{ k } ,\] then a unit vector normal to the vectors \[\vec{a} + \vec{b} \text{ and } \vec{b} - \vec{c}\] is
If \[\vec{a} = 2 \hat{ i } - 3 \hat{ j } - \hat{ k } \text{ and } \vec{b} = \hat{ i } + 4 \hat{ j } - 2 \hat{ k
} , \text{ then } \vec{a} \times \vec{b}\] is
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 \[\left( \vec{a} \times \vec{b} \right)^2\] 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 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-
Let `hata` and `hatb` be two unit vectors such that the angle between them is `π/4`. If θ is the angle between the vectors `(hata + hatb)` and `(hata xx 2hatb + 2(hata xx hatb))`, then the value of 164 cos2θ is equal to ______.
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`.
