Advertisements
Advertisements
प्रश्न
Find a unit vector perpendicular to both the vectors \[4 \hat{ i } - \hat{ j } + 3 \hat{ k } \text{ and } - 2 \hat{ i } + \hat{ j } - 2 \hat{ k } .\]
उत्तर
\[ \text{ Given } : \]
\[ \vec{a} = 4 \hat { i } - \hat{ j } + 3 \hat{ k } \]
\[ \vec{b} = - 2 \hat{ i } + \hat{ j } - 2 \hat{ k } \]
\[ \therefore \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k} \\ 4 & - 1 & 3 \\ - 2 & 1 & - 2\end{vmatrix}\]
\[ = \left( 2 - 3 \right) \hat{ i } - \left( - 8 + 6 \right) \hat{ j } + \left( 4 - 2 \right) \hat{ k } \]
\[ = - \hat{ i } + 2 \hat{ j } + 2 \hat{ k } \]
\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = \sqrt{1 + 2^2 + 2^2}\]
\[ = \sqrt{9}\]
\[ = 3\]
\[ \text{ Unit vector perpendicular to } \vec{a} \text{ and } \vec{b} =\frac{\vec{a} \times \vec{b}}{\left| \vec{a} \times \vec{b} \right|} = \frac{- \hat{ i } + 2 \hat{ j } + 2 \hat{ k } }{3}\]
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.
If either `veca = vec0` or `vecb = vec0`, then `veca xxvecb = vec0`. Is the converse true? Justify your answer with an example.
Find the area of the parallelogram whose adjacent sides are determined by the vector `veca = hati - hatj + 3hatk` and `vecb = 2hati - 7hatj + hatk`.
\[\text{ If } \vec{a} = \hat { i } + 3 \hat { j } - 2 \hat { k } \text{ and } \vec{b} = - \hat { i } + 3 \hat { k } , \text{ find } \left| \vec{a} \times \vec{b} \right| .\]
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 a vector of magnitude 49, which is perpendicular to both the vectors \[2 \hat{ i } + 3 \hat{ j } + 6 \hat{ k } \text{ and } 3 \hat{ i } - 6 \hat{ j } + 2 \hat{ k } .\]
Find the area of the parallelogram determined by the vector \[2 \hat{ i } + \hat{ j } + 3 \hat{ k } \text{ and } \hat{ i } - \hat{ j } \] .
Find the area of the parallelogram whose diagonals are \[3 \hat{ i } + 4 \hat{ j } \text{ and } \hat{ i } + \hat{ j } + \hat{ k }\]
Find the angle between two vectors \[\vec{a} \text{ and } \vec{b}\] , if \[\left| \vec{a} \times \vec{b} \right| = \vec{a} \cdot \vec{b} .\]
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.
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 \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} .\]
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 \[\vec{a} \text{ and } \vec{b}\] .
If \[\vec{a} = 2 \hat{ i } - 3 \hat{ j } + \hat{ k } , \vec{b} = -\hat{ i } + \hat{ k } , \vec{c} = 2 \hat{ j } - \hat{ k } \] are three vectors, find the area of the parallelogram having diagonals \[\left( \vec{a} + \vec{b} \right)\] and \[\left( \vec{b} + \vec{c} \right)\] .
Using vectors, find the area of the triangle with vertice A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1) .
If \[\left| \vec{a} \times \vec{b} \right|^2 + \left| \vec{a} \cdot \vec{b} \right|^2 = 400\] and \[\left| \vec{a} \right| = 5,\] then write the value of \[\left| \vec{b} \right| .\]
Define vector product of two vectors.
Write the value of \[\hat{ i } × \left( \hat{ j } + \hat{ k } \right) + \hat{ j } × \left( \hat{ k } + \hat{ i } \right) + \hat{ k } × \left( \hat{ i } + \hat{ j } \right) .\]
For any two vectors \[\vec{a}\] and \[\vec{b}\] , find \[\vec{a} . \left( \vec{b} \times \vec{a} \right) .\]
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 angle between the vectors \[\vec{a} \times \vec{b}\] and \[\vec{b} \times \vec{a}\] .
If \[\hat{ i } , \hat{ j } , \hat{ k } \] are unit vectors, then
The value of \[\left( \vec{a} \times \vec{b} \right)^2\] 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 ______.
What is the sum of vector `veca = hati - 2hati + hatk, vecb = - 2hati + 4hatj + 5hatk` and `vecc = hati - 6hatj - 7hatk`
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 `veca` and `vecb` are unit vectors inclined at an angle 30° to each other, then find the area of the parallelogram with `(veca + 3vecb)` and `(3veca + vecb)` as adjacent sides.
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 `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 ______.
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 the angle between `veca` and `vecb` is `π/3` and `|veca xx vecb| = 3sqrt(3)`, then the value of `veca.vecb` is ______.
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`.
