Advertisements
Advertisements
प्रश्न
If a, b, c are the lengths of sides, BC, CA and AB of a triangle ABC, prove that \[\vec{BC} + \vec{CA} + \vec{AB} = \vec{0}\] and deduce that \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} .\]
Advertisements
उत्तर
\[\text{ We have } \]
\[ \vec{BC} = \vec{a} \]
\[ \vec{CA} = \vec{b} \]
\[ \vec{AB} = \vec{c} \]
\[\left| \vec{a} \right|=a\]
\[\left| \vec{b} \right| =b ( \because \text{ Length is always positive} )\]
\[ \vec{c} =c \]
\[ \text{ Now } , \]
\[ \vec{BC} + \vec{CA} + \vec{AB} = \vec{0} ( \text{ Given } )\]
\[ \Rightarrow \vec{a} + \vec{b} + \vec{c} = \vec{0} \]
\[ \Rightarrow \vec{a} \times \left( \vec{a} + \vec{b} + \vec{c} \right) = \vec{a} \times \vec{0} \]
\[ \Rightarrow \vec{a} \times \vec{a} + \vec{a} \times \vec{b} + \vec{a} \times \vec{c} = \vec{0} \]
\[ \Rightarrow \vec{0} + \vec{a} \times \vec{b} - \vec{c} \times \vec{a} = \vec{0} \]
\[ \Rightarrow \vec{a} \times \vec{b} = \vec{c} \times \vec{a} \]
\[ \Rightarrow \left| \vec{a} \right| \left| \vec{b} \right|\sin C = \left| \vec{c} \right| \left| \vec{a} \right| \sin B\]
\[ \Rightarrow ab \sin C = ca \sin B\]
\[\text{ Dividing both sides by abc,we get } \]
\[ \Rightarrow \frac{\sin C}{c} = \frac{\sin B}{b} . . . (1)\]
\[\text{ Again } ,\]
\[ \vec{BC} + \vec{CA} + \vec{AB} = \vec{0} \]
\[ \Rightarrow \vec{a} + \vec{b} + \vec{c} = \vec{0} \]
\[ \Rightarrow \vec{b} \times \left( \vec{a} + \vec{b} + \vec{c} \right) = \vec{b} \times \vec{0} \]
\[ \Rightarrow \vec{b} \times \vec{a} + \vec{b} \times \vec{b} + \vec{b} \times \vec{c} = \vec{0} \]
\[ \Rightarrow - \vec{a} \times \vec{b} + \vec{0} + \vec{b} \times \vec{c} = \vec{0} \]
\[ \Rightarrow \vec{a} \times \vec{b} = \vec{b} \times \vec{c} \]
\[ \Rightarrow \left| \vec{a} \right| \left| \vec{b} \right| \sin C = \left| \vec{b} \right| \left| \vec{c} \right| \sin A\]
\[ \Rightarrow ab \sin C = bc \sin A\]
\[\text{ Dividing both sides by abc,we get } \]
\[ \Rightarrow \frac{\sin C}{c} = \frac{\sin A}{a} . . . (2)\]
\[\text{ From (1) and (2), we get } \]
\[\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\]
\[ \Rightarrow \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]
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`.
Find λ and μ if `(2hati + 6hatj + 27hatk) xx (hati + lambdahatj + muhatk) = vec0`.
Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).
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 \[\vec{a} = \hat{i}+ \hat{j} + \hat{k} , \vec{b} =\hat {i} + 2 \hat{j} + 3 \hat{k}\].
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 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 determined by the vector \[2 \hat{ i } + \hat{ j } + 3 \hat{ k } \text{ and } \hat{ i } - \hat{ j } \] .
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 \[2 \hat{ i } + 3 \hat{ j } + 6 \hat{ k } \text{ and } 3 \hat{ i } - 6 \hat{ j } + 2 \hat{ k } \]
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.
For any two vectors \[\vec{a} \text{ and } \vec{b}\] , prove that \[\left| \vec{a} \times \vec{b} \right|^2 = \begin{vmatrix}\vec{a} . \vec{a} & & \vec{a} . \vec{b} \\ \vec{b} . \vec{a} & & \vec{b} . \vec{b}\end{vmatrix}\]
If \[\vec{a} = a_1 \hat{ i } + a_2 \hat{ j } + a_3 \hat{ k } , \vec{b} = b_1 \hat{ i } + b_2 \hat{ j } + b_3 \hat{ k } \text{ and } \vec{c} = c_1 \hat{ i } + c_2 \hat{ j } + c_3 \hat{ k } ,\]then verify that \[\vec{a} \times \left( \vec{b} + \vec{c} \right) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c} .\]
Find all vectors of magnitude \[10\sqrt{3}\] that are perpendicular to the plane of \[\hat{ i } + 2 \hat{ j } + \hat{ k } \] and \[- \hat { i } + 3 \hat{ j } + 4 \hat{ k } \] .
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 \[\vec{a} \text{ and } \vec{b} .\]
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 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 \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left| \vec{a} . \vec{b} \right| = \left| \vec{a} \times \vec{b} \right|,\] write the angle between \[\vec{a} \text{ and } \vec{b} .\]
If \[\vec{a}\] is a unit vector such that \[\vec{a} \times \hat{ i } = \hat{ j } , \text{ find } \vec{a} . \hat{ i } \] .
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 \[\vec{a} = 2 \hat{ i } + \hat{ j } + 2 \hat{ k } \text{ and } \vec{b} = \hat{ j } + \hat{ k } \] .
Write the angle between the vectors \[\vec{a} \times \vec{b}\] and \[\vec{b} \times \vec{a}\] .
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
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
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
What is the sum of vector `veca = hati - 2hati + hatk, vecb = - 2hati + 4hatj + 5hatk` and `vecc = hati - 6hatj - 7hatk`
Let `veca = hati + hatj, vecb = hati - hatj` and `vecc = hati + hatj + hatk`. If `hatn` is a unit vector such that `veca.hatn` = 0 and `vecb.hatn` = 0, then find `|vecc.hatn|`.
Let `veca = 2hati + hatj - 2hatk, vecb = hati + hatj`. If `vecc` is a vector such that `veca . vecc = \|vecc|, |vecc - veca| = 2sqrt(2)` and the angle between `veca xx vecb` and `vecc` is 30°, then `|(veca xx vecb) xx vecc|` equals ______.
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 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|`.
