Advertisements
Advertisements
प्रश्न
If \[\vec{a}\] is any vector, then \[\left( \vec{a} \times \hat{ i } \right)^2 + \left( \vec{a} \times \hat{ j } \right)^2 + \left( \vec{a} \times \hat{ k } \right)^2 =\]
पर्याय
- \[\vec{a}^2\]
\[2 \vec{a}^2\]
- \[3 \vec{a}^2\]
\[4 \vec{a}^2\]
उत्तर
\[2 \vec{a}^2\]
\[\text{ Let } \vec{a} = a_1 \hat{ i } + a_2 \hat{ j } + a_3 \hat{ k } \]
\[ \vec{a} \times \hat{ i } = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ 1 & 0 & 0\end{vmatrix}\]
\[ = a_3 \hat{ j } - a_2 \hat{ k } \]
\[ \Rightarrow \left( \vec{a} \times \hat{ i } \right)^2 = \left( a_3 \hat{ j } - a_2 \hat{ k } \right)^2 \]
\[ = {a_3}^2 \left| \hat{ j } \right|^2 + {a_2}^2 \left| \hat{ k } \right|^2 - 2 a_3 a_2 \left( \hat{ j } . \hat{ k } \right)\]
\[ = {a_3}^2 + {a_2}^2 (\because \hat{ j } . \hat{ k } =0) . . . (1)\]
\[ \therefore \vec{a} \times \hat{ j } = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ 0 & 1 & 0\end{vmatrix}\]
\[ = - a_3 \hat{ i } + a_1 \hat{ k } \]
\[ \Rightarrow \left( \vec{a} \times \hat{ j } \right)^2 = \left( - a_3 \hat{ i } + a_1 \hat{ k } \right)^2 \]
\[ = {a_3}^2 \left| \hat{ i } \right|^2 + {a_1}^2 \left| \hat{ k } \right|^2 - 2 a_3 a_2 \left( \hat{ i} . \hat{ k } \right)\]
\[ = {a_3}^2 + {a_1}^2 (\because \hat{ i } .\hat{ k } =0) . . . (2)\]
\[ \therefore \vec{a} \times \hat{ k } = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ 0 & 0 & 1\end{vmatrix}\]
\[ = a_2 \hat{ i } - a_1 \hat{ j } \]
\[ \Rightarrow \left( \vec{a} \times k \right)^2 = \left( a_2 \hat{ i} - a_1 \hat{ j} \right)^2 \]
\[ = {a_2}^2 \left| \hat{ i } \right|^2 + {a_1}^2 \left| j \right|^2 + 2 a_1 a_2 \left( \hat{ i } . \hat{ j },\right)\]
\[ = {a_2}^2 + {a_1}^2 (\because \hat{ i } . \hat{ j } =0) . . . (3)\]
\[\text{ Adding (1), (2) and (3), we get } \]
\[ \left( \vec{a} \times \hat{ i } \right)^2 + \left( \vec{a} \times \hat{ j } \right)^2 + \left( \vec{a} \times k \right)^2 = {a_3}^2 + {a_2}^2 + {a_3}^2 + {a_1}^2 + {a_2}^2 + {a_1}^2 \]
\[ = 2 \left( {a_1}^2 + {a_2}^2 + {a_3}^2 \right)\]
\[ = 2 \vec{a}^2 (\because\left| \vec{a} \right|=\sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2})\]
APPEARS IN
संबंधित प्रश्न
Find `|veca × vecb|`, if `veca = hati - 7hatj + 7hatk` and `vecb = 3hati - 2hatj + 2hatk`.
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`.
Let the vectors `veca, vecb, vecc` given as `a_1hati + a_2hatj + a_3hatk, b_1hati + b_2hatj + b_3hatk, c_1hati + c_2hatj + c_3hatk` Then show that = `veca xx (vecb+ vecc) = veca xx vecb + veca xx vecc.`
If either `veca = vec0` or `vecb = vec0`, then `veca xxvecb = vec0`. Is the converse true? Justify your answer with an example.
\[\text{ If } \vec{a} = \hat { i } + 3 \hat { j } - 2 \hat { k } \text{ and } \vec{b} = - \hat { i } + 3 \hat { k } , \text{ find } \left| \vec{a} \times \vec{b} \right| .\]
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 \[3 \hat{ i } + \hat{ j } - 2 \hat{ k } \text{ and } \hat{ i } - 3 \hat{ j } + 4 \hat{ k } \] .
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 \[\vec{p} \text{ and } \vec{q}\] 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.
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 \[\vec{a} \text{ and } \vec{d}\] \[\text{ and } \vec{c} \cdot \vec{d} = 15 .\]
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, 1, 2), B(2, 3, 5) and C(1, 5, 5) .
Find all vectors of magnitude \[10\sqrt{3}\] that are perpendicular to the plane of \[\hat{ i } + 2 \hat{ j } + \hat{ k } \] and \[- \hat { i } + 3 \hat{ j } + 4 \hat{ k } \] .
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.
For any two vectors \[\vec{a} \text{ and } \vec{b} , \text{ find } \left( \vec{a} \times \vec{b} \right) . \vec{b} .\]
If \[\vec{a} = 3 \hat{ i } - \hat{ j } + 2 \hat{ k } \] and \[\vec{b} = 2 \hat { i } + \hat{ j } - \hat{ k} ,\] then find \[\left( \vec{a} \times \vec{b} \right) \vec{a} .\]
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, then write the value of \[\left| \vec{a} \times \vec{b} \right|^2 + \left( \vec{a} . \vec{b} \right)^2 .\]
Find the angle between two vectors \[\vec{a} \text{ and } \vec{b}\] with magnitudes 1 and 2 respectively and when \[\left| \vec{a} \times \vec{b} \right| = \sqrt{3} .\]
Find a vector of magnitude \[\sqrt{171}\] which is perpendicular to both of the vectors \[\vec{a} = \hat{ i } + 2 \hat{ j } - 3 \hat{ k } \] and \[\vec{a} = \hat{ i } + 2 \hat{ j } - 3 \hat{ k } \] .
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
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
Vectors \[\vec{a} \text{ and } \vec{b}\] are inclined at angle θ = 120°. If \[\left| \vec{a} \right| = 1, \left| \vec{b} \right| = 2,\] then \[\left[ \left( \vec{a} + 3 \vec{b} \right) \times \left( 3 \vec{a} - \vec{b} \right) \right]^2\] is equal to
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 \[\left| \vec{a} \times \vec{b} \right| = 4, \left| \vec{a} \cdot \vec{b} \right| = 2, \text{ then } \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 =\]
The value of \[\left( \vec{a} \times \vec{b} \right)^2\] is
The value of λ for which the two vectors `2hati - hatj + 2hatk` and `3hati + λhatj + hatk` are perpendicular is ______.
Let `veca` and `vecb` be two unit vectors and θ is the angle between them, Then `veca + vecb` is a unit vector if-
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 `veca = hati + hatj + hatk` and `vecb = hati + 2hatj + 3hatk` then find a unit vector perpendicular to both `veca + vecb` and `veca - vecb`.
Find the area of the parallelogram whose diagonals are `hati - 3hatj + hatk` and `hati + hatj + hatk`.
