Advertisements
Advertisements
Question
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.
Advertisements
Solution
\[\text{ Given } : \]
\[ \vec{a} = \frac{1}{7} \left( 2 \hat{ i } + 3 \hat{ j }+ 6 \hat{ k } \right)\]
\[ \vec{b} = \frac{1}{7} \left( 3 \hat{ i } - 6 \hat{ j } + 2 \hat { k } \right)\]
\[ \vec{c} = \frac{1}{7}\left( 6 \hat{ i } + 2 \hat{ j } - 3 \hat{ k} \right)\]
\[ \vec{a} \times \vec{b} = \left( \frac{1}{7} \right) \left( \frac{1}{7} \right)\begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & 3 & 6 \\ 3 & - 6 & 2\end{vmatrix}\]
\[ = \frac{1}{49}\left( 42 \hat{ i } + 14 \hat{ j } - 21 \hat{ k } \right)\]
\[ = \frac{1}{49}\left[ 7 \left( 6 \hat{ i } + 2 \hat{ j } - 3 \hat{ k } \right) \right]\]
\[ = \frac{1}{7}\left( 6 \hat{ i } + 2 \hat{ j } - 3 \hat{ k } \right)\]
\[ = \vec{c} \]
\[ \vec{b} \times \vec{c} = \left( \frac{1}{7} \right) \left( \frac{1}{7} \right)\begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 3 & - 6 & 2 \\ 6 & 2 & - 3\end{vmatrix}\]
\[ = \frac{1}{49}\left( 14 \hat{ i } + 21 \hat{ j } + 42 \hat{ k } \right)\]
\[ = \frac{1}{49}\left[ 7 \left( 2 \hat{ i } + 3 \hat{ j} + 6 \hat{ k } \right) \right]\]
\[ = \frac{1}{7} \left( 2 \hat{ i } + 3 \hat{ j } + 6 \hat{ k } \right)\]
\[ = \vec{a} \]
\[ \vec{c} \times \vec{a} = \left( \frac{1}{7} \right) \left( \frac{1}{7} \right)\begin{vmatrix}\hat{ i } & \hat{ j } & k \\ 6 & 2 & - 3 \\ 2 & 3 & 6\end{vmatrix}\]
\[ = \frac{1}{49}\left( 21 \hat{ i } - 42 \hat{ j } + 14 \hat{ k } \right)\]
\[ = \frac{1}{49}\left[ 7 \left( 3 \hat{ i } - 6 \hat{ j } + 2 \hat{ k } \right) \right]\]
\[ = \frac{1}{7} \left( 3 \hat{ i } - 6 \hat{ j } + 2 \hat { k } \right)\]
\[ = \vec{b} \]
\[\left| \vec{a} \right| = \frac{1}{7}\sqrt{4 + 9 + 36}\]
\[ = \frac{7}{7}\]
\[ = 1\]
\[\left| \vec{b} \right| = \frac{1}{7}\sqrt{9 + 36 + 4}\]
\[ = \frac{7}{7}\]
\[ = 1\]
\[\left| \vec{c} \right| = \frac{1}{7}\sqrt{36 + 4 + 9}\]
\[ = \frac{7}{7}\]
\[ = 1\]
\[\text{ Thus } , \vec{a} , \vec{b} \text{ and } \vec{c} \text{ form a right handed orthogonal system of unit vectors. } \]
APPEARS IN
RELATED QUESTIONS
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.
Find `|veca × vecb|`, if `veca = hati - 7hatj + 7hatk` and `vecb = 3hati - 2hatj + 2hatk`.
Show that `(veca - vecb) xx (veca + vecb) = 2(veca xx vecb)`.
Given that `veca.vecb = 0` and `veca xx vecb = 0` What can you conclude about the vectors `veca and vecb`?
Let the vectors `veca` and `vecb` be such that `|veca| = 3` and `|vecb| = sqrt2/3`, then `veca xx vecb` is a unit vector, if the angle between `veca` and `vecb` is ______.
Find a unit vector perpendicular to both the vectors \[\vec{a} + \vec{b} \text { and } \vec{a} - \vec{b}\] ,where \[\vec{a} = \hat{i}+ \hat{j} + \hat{k} , \vec{b} =\hat {i} + 2 \hat{j} + 3 \hat{k}\].
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 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 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 \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 7 \text{ and } \vec{a} \times \vec{b} = 3 \hat{ i } + 2 \hat{ j } + 6 \hat{ k } ,\] 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.
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 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 \[- \hat { i } + 3 \hat{ j } + 4 \hat{ k } \] .
Define vector product of two vectors.
Write the value of \[\hat{ i } . \left( \hat{ j } \times \hat{ k } \right) + \hat{ j } . \left( \hat{ k } \times \hat{ i } \right) + \hat{ k } . \left( \hat{ i } \times \hat{ j } \right) .\]
For any two vectors \[\vec{a}\] and \[\vec{b}\] , find \[\vec{a} . \left( \vec{b} \times \vec{a} \right) .\]
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 \[\vec{a} \text{ and } \vec{b}\] are unit vectors, then write the value of \[\left| \vec{a} \times \vec{b} \right|^2 + \left( \vec{a} . \vec{b} \right)^2 .\]
Find 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 \[\vec{a} = \hat{ i } + 2 \hat{ j } - 3 \hat{ k } \] .
Write the number of vectors of unit length perpendicular to both the vectors \[\vec{a} = 2 \hat{ i } + \hat{ j } + 2 \hat{ k } \text{ and } \vec{b} = \hat{ j } + \hat{ k } \] .
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
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
The value of \[\left( \vec{a} \times \vec{b} \right)^2\] is
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 ______.
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 `veca, vecb, vecc` be three vectors mutually perpendicular to each other and have same magnitude. If a vector `vecr` satisfies. `veca xx {(vecr - vecb) xx veca} + vecb xx {(vecr - vecc) xx vecb} + vecc xx {(vecr - veca) xx vecc} = vec0`, then `vecr` 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`.
If `veca xx vecb = veca xx vecc` where `veca, vecb` and `vecc` are non-zero vectors, then prove that either `vecb = vecc` or `veca` and `(vecb - vecc)` are parallel.
If `veca` is a unit vector perpendicular to `vecb` and `(veca + 2vecb).(3veca - vecb) = -5`, find `|vecb|`.
