Advertisements
Advertisements
प्रश्न
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
पर्याय
\[2 \hat{ i } + \hat{ j } + \hat{ k } \]
\[\sqrt{6}\left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)\]
\[\frac{1}{\sqrt{6}}\left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)\]
\[\frac{1}{6}\left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)\]
उत्तर
\[\frac{1}{\sqrt{6}}\left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)\]
\[\text{ The vector } \vec{PQ} \times \vec{PR} \text{ is perpendicular to the vectors } \vec{PQ} \text{ and } \vec{PR} . \]
\[ \therefore \text{ Required unit vector} = \frac{\vec{PQ} \times \vec{PR}}{\left| \vec{PQ} \times \vec{PR} \right|}\]
\[\text{ Now } , \]
\[ \vec{PQ} = P . V . \text{ of } Q - P . V . of P\]
\[ = \hat{ i } + \hat{ j } - 3 \hat{ k } \]
\[ \vec{PR} = P . V . \text{ of } R - P . V . of P\]
\[ = - \hat{ i } + 3 \hat{ j } - \hat{ k } \]
\[ \therefore \vec{PQ} \times \vec{PR} = \begin{vmatrix}\text{ i } & \text{ j } & \text{ k } \\ 1 & 1 & - 3 \\ - 1 & 3 & - 1\end{vmatrix}\]
\[ = 8 \hat{ i } + 4 \hat{ j } + 4 \text{ k } \]
\[ = 4 \left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)\]
\[ \Rightarrow \left| \vec{PQ} \times \vec{PR} \right| = \sqrt{64 + 16 + 16}\]
\[ = \sqrt{96}\]
\[ = 4\sqrt{6}\]
\[\text{ Required unit vector } = \frac{\vec{PQ} \times \vec{PR}}{\left| \vec{PQ} \times \vec{PR} \right|}\]
\[ = \frac{4 \left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)}{4\sqrt{6}}\]
\[ = \frac{1}{\sqrt{6}}\left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)\]
APPEARS IN
संबंधित प्रश्न
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`.
If either `veca = vec0` or `vecb = vec0`, then `veca xxvecb = vec0`. Is the converse true? Justify your answer with an example.
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 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 \[3 \hat{ i } + \hat{ j } - 2 \hat{ k } \text{ and } \hat{ i } - 3 \hat{ j } + 4 \hat{ k } \] .
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 \[2 \hat{ i }+ \hat{ k } \text{ and } \hat{ i } + \hat{ j } + \hat{ k } \]
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 \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 7 \text{ and } \vec{a} \times \vec{b} = 3 \hat{ i } + 2 \hat{ j } + 6 \hat{ k } ,\] find the angle between \[\vec{a} \text{ and } \vec{b} .\]
If \[\vec{a,} \vec{b,} \vec{c}\] are three unit vectors such that \[\vec{a} \times \vec{b} = \vec{c} , \vec{b} \times \vec{c} = \vec{a,} \vec{c} \times \vec{a} = \vec{b} .\] Show that \[\vec{a,} \vec{b,} \vec{c}\] form an orthonormal right handed triad of unit vectors.
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.
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 \[\vec{a} \text{ and } \vec{d}\] \[\text{ and } \vec{c} \cdot \vec{d} = 15 .\]
Using vectors find the area of the triangle with vertices, A (2, 3, 5), B (3, 5, 8) and C (2, 7, 8).
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.
Using vectors, find the area of the triangle with vertice A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1) .
Find all vectors of magnitude \[10\sqrt{3}\] that are perpendicular to the plane of \[\hat{ i } + 2 \hat{ j } + \hat{ k } \] and \[- \hat { i } + 3 \hat{ j } + 4 \hat{ k } \] .
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 two vectors such that \[\left| \vec{a} \times \vec{b} \right| = \sqrt{3}\text{ and } \vec{a} . \vec{b} = 1,\] find the angle between.
If \[\vec{a} = 3 \hat{ i } - \hat{ j } + 2 \hat{ k } \] and \[\vec{b} = 2 \hat { i } + \hat{ j } - \hat{ k} ,\] then find \[\left( \vec{a} \times \vec{b} \right) \vec{a} .\]
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 \[\left| \vec{r} \times \hat{ i } \right|^2 .\]
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 \[\vec{a} \text{ and } \vec{b}\] .
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 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 \[\vec{a,} \vec{b}\] represent the diagonals of a rhombus, then
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 \[\left| \vec{a} \times \vec{b} \right| = 4, \left| \vec{a} \cdot \vec{b} \right| = 2, \text{ then } \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 =\]
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.
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 ______.
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 ______.
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`.
