Advertisements
Advertisements
प्रश्न
The two adjacent sides of a parallelogram are \[2 \hat{ i } - 4 \hat{ j } + 5 \hat{ k } \text{ and } \hat{ i } - 2 \hat{ j } - 3\hat{ k } .\]\ Find the unit vector parallel to one of its diagonals. Also, find its area.
Advertisements
उत्तर
\[\text{ Suppose } \square ABCD \text{ is the given parallelogram and AC is its diagonal } . \]
\[\text{ Let } : \]
\[ \vec{AB} = 2 \hat{ i } - 4 \hat{ j } + 5 \hat{ k } \]
\[ \vec{BC} = \hat{ i } - 2 \hat{ j } - 3 \hat{ k } \]
\[ \therefore \text{ Diagonal } \vec{AC} = \vec{AB} + \vec{BC} \]
\[ = 3 \hat{ i } - 6 \hat{ j } + 2 \hat{ k } \]
\[ \Rightarrow \left| \vec{AC} \right| = \sqrt{9 + 36 + 4}\]
\[ = 7\]
\[\text{ Unit vector parallel to } \vec{AC} =\frac{\vec{AC}}{\left| \vec{AC} \right|}\]
\[ =\frac{3 \hat{ i } - 6 \hat{ j } + 2 \hat{ k } }{7}\]
\[\text{ Now } ,\]
\[ \vec{AB} \times \vec{BC} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & - 4 & 5 \\ 1 & - 2 & - 3\end{vmatrix}\]
\[ = 22 \hat{ i } + 11 \hat{ j } + 0 \hat{ k } \]
\[ \Rightarrow \left| \vec{AB} \times \vec{AC} \right| = \sqrt{484 + 121}\]
\[ = \sqrt{605}\]
\[ = 11\sqrt{5}\]
\[Area of triangleABC=\frac{1}{2}\left| \vec{AB} \times \vec{AC} \right|\]
\[ = \frac{11\sqrt{5}}{2}\text{ sq . units } \]
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`.
Show that `(veca - vecb) xx (veca + vecb) = 2(veca xx vecb)`.
Find λ and μ if `(2hati + 6hatj + 27hatk) xx (hati + lambdahatj + muhatk) = vec0`.
Given that `veca.vecb = 0` and `veca xx vecb = 0` What can you conclude about the vectors `veca and vecb`?
If either `veca = vec0` or `vecb = vec0`, then `veca xxvecb = vec0`. Is the converse true? Justify your answer with an example.
If θ is the angle between two vectors `hati - 2hatj + 3hatk and 3hati - 2hatj + hatk` find `sin theta`
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`
If A, B, C are three non- collinear points with position vectors `vec a, vec b, vec c`, respectively, then show that the length of the perpendicular from Con AB is `|(vec a xx vec b)+(vec b xx vec c) + (vec b xx vec a)|/|(vec b - vec a)|`
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 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 \[2 \hat{ i } \text{ and } 3 \hat{ j } \] .
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 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 \[3 \hat{ i } + 4 \hat{ j } \text{ and } \hat{ i } + \hat{ j } + \hat{ k }\]
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.
if \[\vec{a} = \hat{ i }- 2\hat{ j } + 3 \hat{ k } , \text{ and } \vec{b} = 2 \hat{ i } + 3 \hat{ j } - 5 \hat{ k } ,\] then find \[\vec{a} \times \vec{b} .\] Verify th at \[\vec{a} \text{ and } \vec{a} \times \vec{b}\] are perpendicular to each other.
Find the area of the triangle formed by O, A, B when \[\vec{OA} = \hat{ i } + 2 \hat{ j } + 3 \hat{ k } , \vec{OB} = - 3 \hat{ i } - 2 \hat{ j }+ \hat{ k } .\]
Using vectors, find the area of the triangle with vertice A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1) .
Define vector product of two vectors.
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 \[\vec{r} = x \hat{ i } + y \hat{ j } + z \hat{ k } ,\] 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 \[\vec{a} \text{ and } \vec{b}\] .
If \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left| \vec{a} . \vec{b} \right| = \left| \vec{a} \times \vec{b} \right|,\] write the angle between \[\vec{a} \text{ and } \vec{b} .\]
Write the value of \[\left( \hat{ i } \times \hat{ j } \right) \cdot \hat{ k } + \left( \hat{ j } + \hat{ k } \right) \cdot \hat{ j } \]
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 `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 `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 xx vecb| = sqrt(3)` and `veca.vecb` = – 3, then angle between `veca` and `vecb` is ______.
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` and `vecb` are two non-zero vectors such that `|veca xx vecb| = veca.vecb`, find the angle between `veca` and `vecb`.
