Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
\[\text{ Given } : \]
\[ \vec{a} = 2 \hat{ i } + 5 \hat{ j } - 7 \hat{ k }\]
\[ \vec{b} = - 3 \hat{ i } + 4 \hat{ j } + \hat{ k } \]
\[ \vec{c} = \hat{ i } - 2 \hat{ j } - 3 \hat{ k } \]
\[ \therefore \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & 5 & - 7 \\ - 3 & 4 & 1\end{vmatrix}\]
\[ = \left( 5 + 28 \right) \hat{ i } - \left( 2 - 21 \right) \hat{ j } + \left( 8 + 15 \right) \hat{ k } \]
\[ = 33 \hat{ i } + 19 \hat{ j }+ 23 \hat{ k } \]
\[ \Rightarrow \left( \vec{a} \times \vec{b} \right) \times \vec{c} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 33 & 19 & 23 \\ 1 & - 2 & - 3\end{vmatrix}\]
\[ = \left( - 57 + 46 \right) \hat{ i } - \left( - 99 - 23 \right) \hat{ j } + \left( - 66 - 19 \right) \hat{ k } \]
\[ \Rightarrow \left( \vec{a} \times \vec{b} \right) \times \vec{c} = - 11 \hat{ i } + 122 \hat{ j } - 85 \hat{ k} . . . (1)\]
\[ \therefore \vec{b} \times \vec{c} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ - 3 & 4 & 1 \\ 1 & - 2 & - 3\end{vmatrix}\]
\[ = \left( - 12 + 2 \right) \hat{ i } - \left( 9 - 1 \right) \hat{ j } + \left( 6 - 4 \right) \hat{ k } \]
\[ = - 10 \hat{ i } - 8 \hat{ j }+ 2 \hat{ k } \]
\[ \Rightarrow \vec{a} \times \left( \vec{b} \times \vec{c} \right) = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat { k } \\ 2 & 5 & - 7 \\ - 10 & - 8 & 2\end{vmatrix}\]
\[ = \left( 10 - 56 \right) \hat{ i } - \left( 4 - 70 \right) \hat{ j } + \left( - 16 + 50 \right) \hat{ k } \]
\[ \Rightarrow \vec{a} \times \left( \vec{b} \times \vec{c} \right) = - 46 \hat{ i } + 66 \hat{ j } + 34 \hat{ k } . . . (2)\]
\[\text{ From (1) and (2), we get } \]
\[\left( \vec{a} \times \vec{b} \right) \times \vec{c} \neq \vec{a} \times \left( \vec{b} \times \vec{c} \right)\]
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 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`.
Show that `(veca - vecb) xx (veca + vecb) = 2(veca xx vecb)`.
Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).
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 a vector whose length is 3 and which is perpendicular to the vector \[\vec{a} = 3 \hat{ i } + \hat{ j } - 4 \hat{ k } \text{ and } \vec{b} = 6 \hat{ i } + 5 \hat{ j } - 2 \hat{ k } .\]
Find the area of the parallelogram whose diagonals are \[3 \hat{ i } + 4 \hat{ j } \text{ and } \hat{ i } + \hat{ j } + \hat{ k }\]
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 \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 7 \text{ and } \vec{a} \times \vec{b} = 3 \hat{ i } + 2 \hat{ j } + 6 \hat{ k } ,\] find the angle between \[\vec{a} \text{ and } \vec{b} .\]
What inference can you draw if \[\vec{a} \times \vec{b} = \vec{0} \text{ and } \vec{a} \cdot \vec{b} = 0 .\]
Find the area of the triangle formed by O, A, B when \[\vec{OA} = \hat{ i } + 2 \hat{ j } + 3 \hat{ k } , \vec{OB} = - 3 \hat{ i } - 2 \hat{ j }+ \hat{ k } .\]
Find a unit vector perpendicular to each of the vectors \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} , \text{ where } \vec{a} = 3 \hat{ i } + 2 \hat{ j } + 2 \hat{ k } \text{ and } \vec{b} = \hat{ i } + 2 \hat{ j } - 2 \hat{ k } .\]
Using vectors, find the area of the triangle with vertice A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5) .
Define vector product of two vectors.
Write the expression for the area of the parallelogram having \[\vec{a} \text{ and } \vec{b}\] as its diagonals.
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 \[\vec{a} \times \left( \vec{b} + \vec{c} \right) + \vec{b} \times \left( \vec{c} + \vec{a} \right) + \vec{c} \times \left( \vec{a} + \vec{b} \right) .\]
If \[\vec{r} = x \hat{ i } + y \hat{ j } + z \hat{ k } ,\] then write the value of \[\left| \vec{r} \times \hat{ i } \right|^2 .\]
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 \[\vec{a,} \vec{b}\] represent the diagonals of a rhombus, then
A unit vector perpendicular to both \[\hat{ i } + \hat{ j } \text{ and } \hat{ j } + \hat{ k } \] 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 \[\hat{ i } \cdot \left( \hat{ j } \times \hat{ k } \right) + \hat{ j } \cdot \left( \hat{ i } \times \hat{ k } \right) + \hat{ k } \cdot \left( \hat{ i } \times \hat{ j } \right),\] is
The value of λ for which the two vectors `2hati - hatj + 2hatk` and `3hati + λhatj + hatk` are perpendicular 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 ______.
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.
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 angle between `veca` and `vecb` is `π/3` and `|veca xx vecb| = 3sqrt(3)`, then the value of `veca.vecb` is ______.
If `veca` and `vecb` are two non-zero vectors such that `|veca xx vecb| = veca.vecb`, find the angle between `veca` and `vecb`.
If `veca` is a unit vector perpendicular to `vecb` and `(veca + 2vecb).(3veca - vecb) = -5`, find `|vecb|`.
