Advertisements
Advertisements
प्रश्न
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 \[\vec{a} \times \left( \vec{b} + \vec{c} \right) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c} .\]
उत्तर
\[\text{ Given } : \]
\[ \vec{a} = a_1 \hat{ i } + a_2 \hat{ j } + a_3 \hat{ k } \]
\[ \vec{b} = b_1 \hat{ i } + b_2 \hat{ j } + b_3 \hat{ k } \]
\[ \vec{c} = c_1 \hat{ i } + c_2\hat{ j } + c_3 \hat{ k } \]
\[ \vec{b} + \vec{c} = \left( b_1 + c_1 \right) \hat{ i } + \left( b_2 + c_2 \right) \hat{ j } + \left( b_3 + c_3 \right) \hat{ k } \]
\[ \therefore \vec{a} \times \left( \vec{b} + c \right) = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ b_1 + c_1 & b_2 + c_2 & b_3 + c_3\end{vmatrix}\]
\[ = \left( a_2 b_3 + a_2 c_3 - a_3 b_2 - a_3 c_2 \right) \hat{ i } - \left( a_1 b_3 + a_1 c_3 - a_3 b_1 - a_3 c_1 \right) \hat{ j } + \left( a_1 b_2 + a_1 c_2 - a_2 b_1 - a_2 c_1 \right) \hat{ k } . . . (1)\]
\[\text{ Now } , \]
\[ \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\end{vmatrix}\]
\[ = \left( a_2 b_3 - a_3 b_2 \right) \hat{ i } - \left( a_1 b_3 - a_3 b_1 \right) \hat{ j } + \left( a_1 b_2 - a_2 b_1 \right) \hat{ k } \]
\[ \vec{a} \times \vec{c} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ c_1 & c_2 & c_3\end{vmatrix}\]
\[ = \left( a_2 c_3 - a_3 c_2 \right) \hat{ i } - \left( a_1 c_3 - a_3 c_1 \right) \hat{ j } + \left( a_1 c_2 - a_2 c_1 \right) \hat{ k } \]
\[ \vec{a} \times \vec{b} + \vec{b} \times \vec{c} = \left( a_2 b_3 + a_2 c_3 - a_3 b_2 - a_3 c_2 \right) \hat{ i } - \left( a_1 b_3 + a_1 c_3 - a_3 b_1 - a_3 c_1 \right) \hat{ j } + \left( a_1 b_2 + a_1 c_2 - a_2 b_1 - a_2 c_1 \right) \hat{ k } . . . (2)\]
\[ \text{ From (1) and (2), we get } \]
\[ \vec{a} \times \left( \vec{b} + c \right) = \vec{a} \times \vec{b} + \vec{b} \times \vec{c}\]
APPEARS IN
संबंधित प्रश्न
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 ______.
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`
Find a unit vector perpendicular to both the vectors \[\vec{a} + \vec{b} \text { and } \vec{a} - \vec{b}\] ,where \[\vec{a} = \hat{i}+ \hat{j} + \hat{k} , \vec{b} =\hat {i} + 2 \hat{j} + 3 \hat{k}\].
Find the magnitude of \[\vec{a} = \left( 3 \hat{ k } + 4 \hat{ j } \right) \times \left( \hat{ i } + \hat{ j } - \hat{ k } \right) .\]
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 \[\vec{p} \text{ and } \vec{q}\] are unit vectors forming an angle of 30°; find the area of the parallelogram having \[\vec{a} = \vec{p} + 2 \vec{q} \text{ and } \vec{b} = 2 \vec{p} + \vec{q}\] as its diagonals.
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).
The two adjacent sides of a parallelogram are \[2 \hat{ i } - 4 \hat{ j } + 5 \hat{ k } \text{ and } \hat{ i } - 2 \hat{ j } - 3\hat{ k } .\]\ Find the unit vector parallel to one of its diagonals. Also, find its area.
Using vectors, find the area of the triangle with vertice A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5) .
Write the value of \[\hat{ i } . \left( \hat{ j } \times \hat{ k } \right) + \hat{ j } . \left( \hat{ k } \times \hat{ i } \right) + \hat{ k } . \left( \hat{ j } \times \hat{ i } \right) .\]
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 \[\vec{a}\] 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 \[\vec{a} \times \left( \vec{b} + \vec{c} \right) + \vec{b} \times \left( \vec{c} + \vec{a} \right) + \vec{c} \times \left( \vec{a} + \vec{b} \right) .\]
For any two vectors \[\vec{a} \text{ and } \vec{b} , \text{ find } \left( \vec{a} \times \vec{b} \right) . \vec{b} .\]
If \[\vec{a} \text{ and } \vec{b}\] are unit vectors such that \[\vec{a} \times \vec{b}\] is also a unit vector, find the angle between \[\vec{a} \text{ and } \vec{b}\] .
If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, then write the value of \[\left| \vec{a} \times \vec{b} \right|^2 + \left( \vec{a} . \vec{b} \right)^2 .\]
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 value of the area of the parallelogram determined by the vectors \[2 \hat{ i } \text{ and } 3 \hat{ j } .\]
Write the angle between the vectors \[\vec{a} \times \vec{b}\] and \[\vec{b} \times \vec{a}\] .
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
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
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 \[\hat{ i } , \hat{ j } , \hat{ k } \] are unit vectors, then
If θ is the angle between the vectors \[2 \hat{ i } - 2 \hat{ j} + 4 \hat{ k } \text{ and } 3 \hat{ i } + \hat { j } + 2 \hat{ k } ,\] then sin θ =
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 ______.
(a) If `veca = hati - 2j + 3veck , vecb = 2hati + 3hatj - 5hatk,` prove that `veca and vecaxxvecb` are perpendicular.
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.
Let `veca = 2hati + hatj - 2hatk, vecb = hati + hatj`. If `vecc` is a vector such that `veca . vecc = \|vecc|, |vecc - veca| = 2sqrt(2)` and the angle between `veca xx vecb` and `vecc` is 30°, then `|(veca xx vecb) xx vecc|` equals ______.
If `|veca xx vecb| = sqrt(3)` and `veca.vecb` = – 3, then angle between `veca` and `vecb` is ______.
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` is a unit vector perpendicular to `vecb` and `(veca + 2vecb).(3veca - vecb) = -5`, find `|vecb|`.
