Advertisements
Advertisements
प्रश्न
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 =
पर्याय
\[\frac{2}{3}\]
\[\frac{2}{\sqrt{7}}\]
\[\frac{\sqrt{2}}{7}\]
\[\sqrt{\frac{2}{7}}\]
उत्तर
\[\frac{2}{\sqrt{7}}\]
\[\text{ Let } :\]
\[ \vec{a} =2 \hat{ i } -2 \hat{ j } +4 \hat{ k } \]
\[ \vec{b} =3 \hat{ i } + \hat{ j } +2 \hat{ k } \]
\[\left| \vec{a} \right| = \sqrt{2^2 + \left( - 2 \right)^2 + 4^2}\]
\[ = \sqrt{4 + 4 + 16}\]
\[ = \sqrt{24}\]
\[ = 2\sqrt{6}\]
\[ \left| \vec{b} \right| = \sqrt{3^2 + 1^2 + 2^2}\]
\[ = \sqrt{9 + 1 + 4}\]
\[ = \sqrt{14}\]
\[ \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & - 2 & 4 \\ 3 & 1 & 2\end{vmatrix}\]
\[ = - 8 \hat{ i } + 8 \hat{ j } + 8 \hat{ k } \]
\[\left| \vec{a} \times \vec{b} \right| = \sqrt{64 + 64 + 64}\]
\[ = \sqrt{192}\]
\[ = 8 \sqrt{3}\]
\[\text{ Let } \theta \text{ be the angle between } \vec{a} \text{ and } \vec{b .} \]
\[\left| \vec{a} \times \vec{b} \right| = \left| \vec{a} \right| \left| \vec{b} \right| \sin \theta\]
\[ \Rightarrow 8 \sqrt{3} = \left( 2\sqrt{6} \right)\left( \sqrt{14} \right) \sin \theta\]
\[ \Rightarrow \sin \theta = \frac{8 \sqrt{3}}{4\sqrt{21}}\]
\[ = \frac{2}{\sqrt{7}}\]
\[ \Rightarrow \theta = \sin^{- 1} \left( \frac{2}{\sqrt{7}} \right)\]
APPEARS IN
संबंधित प्रश्न
Find `|veca × vecb|`, if `veca = hati - 7hatj + 7hatk` and `vecb = 3hati - 2hatj + 2hatk`.
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`.
Given that `veca.vecb = 0` and `veca xx vecb = 0` What can you conclude about the vectors `veca and vecb`?
If θ is the angle between two vectors `hati - 2hatj + 3hatk and 3hati - 2hatj + hatk` find `sin theta`
Find a unit vector perpendicular to both the vectors \[\vec{a} + \vec{b} \text { and } \vec{a} - \vec{b}\] ,where \[\vec{a} = \hat{i}+ \hat{j} + \hat{k} , \vec{b} =\hat {i} + 2 \hat{j} + 3 \hat{k}\].
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 magnitude of \[\vec{a} = \left( 3 \hat{ k } + 4 \hat{ j } \right) \times \left( \hat{ i } + \hat{ j } - \hat{ k } \right) .\]
Find the area of the parallelogram determined by the vector \[2 \hat{ i } \text{ and } 3 \hat{ j } \] .
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 the area of the parallelogram whose diagonals are \[2 \hat{ i } + 3 \hat{ j } + 6 \hat{ k } \text{ and } 3 \hat{ i } - 6 \hat{ j } + 2 \hat{ k } \]
If \[\vec{a} = 2 \hat{ i } + 5 \hat{ j } - 7 \hat{ k } , \vec{b} = - 3 \hat{ i } + 4 \hat{ j } + \hat{ k } \text{ and } \vec{c} = \hat{ i } - 2 \hat{ j } - 3 \hat{ k } ,\] compute \[\left( \vec{a} \times \vec{b} \right) \times \vec{c} \text{ and } \vec{a} \times \left( \vec{b} \times \vec{c} \right)\] and verify that these are not equal.
Given \[\vec{a} = \frac{1}{7}\left( 2 \hat{ i } + 3 \hat{ j } + 6 \hat{ k } \right), \vec{b} = \frac{1}{7}\left( 3 \hat{ i } - 6 \hat{ j } + 2 \hat{ k } \right), \vec{c} = \frac{1}{7}\left( 6 \hat{ i } + 2 \hat{ j } - 3 \hat{ k }\right), \hat{ i } , \hat{ j } , \hat{ k } \] being a right handed orthogonal system of unit vectors in space, show that \[\vec{a} , \vec{b} , \vec{c}\] is also another system.
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.
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.
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 \[\vec{a} \text{ and } \vec{d}\] \[\text{ and } \vec{c} \cdot \vec{d} = 15 .\]
Using vectors find the area of the triangle with vertices, A (2, 3, 5), B (3, 5, 8) and C (2, 7, 8).
Using vectors, find the area of the triangle with vertice A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5) .
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 \[\left| \vec{b} \right| .\]
Write the value \[\left( \hat{ i } \times \hat{ j } \right) \cdot \hat{ k } + \hat{ i } \cdot \hat{ j } .\]
If \[\vec{a} \text{ and } \vec{b}\] are two vectors of magnitudes 3 and \[\frac{\sqrt{2}}{3}\] espectively such that \[\vec{a} \times \vec{b}\] is a unit vector. Write the angle between \[\vec{a} \text{ and } \vec{b} .\]
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 \[\left| \vec{a} \times \vec{b} \right|^2 + \left( \vec{a} . \vec{b} \right)^2 = 144\] and \[\left| \vec{a} \right| = 4,\] find \[\left| \vec{b} \right|\] .
If \[\vec{r} = x \hat{ i } + y \hat{ j } + z \hat{ k } ,\] then write the value of \[\left| \vec{r} \times \hat{ i } \right|^2 .\]
If \[\vec{c}\] is a unit vector perpendicular to the vectors \[\vec{a} \text{ and } \vec{b} ,\] write another unit vector perpendicular to \[\vec{a} \text{ and } \vec{b} .\]
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 \[\hat{ i } , \hat{ j } , \hat{ k } \] are unit vectors, then
(a) If `veca = hati - 2j + 3veck , vecb = 2hati + 3hatj - 5hatk,` prove that `veca and vecaxxvecb` are perpendicular.
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 `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 ______.
Find the area of the parallelogram whose diagonals are `hati - 3hatj + hatk` and `hati + hatj + hatk`.
