Advertisements
Advertisements
प्रश्न
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|`.
उत्तर
Given, `veca = hati + hatj, vecb = hati - hatj`
and `vecc = hati + hatj + hatk`
Also, given `veca.hatn` = 0
and `vecb.hatn` = 0
Here, `hatn = (veca xx vecb)/(|veca xx vecb|)`
Here, `veca xx vecb = |(hati, hatj, hatk),(1, 1, 0),(1, -1, 0)|`
= `hati(0 - 0) - hatj(0 - 0) + hatk(-1 - 1)`
= `-2hatk`
∴ `hatn = (-2hatk)/sqrt((-2)^2) = - hatk`
Therefore, `|vecc.hatn| = |(hati + hatj + hatk).(-hatk)|`
= |–1|
= 1
APPEARS IN
संबंधित प्रश्न
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`?
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.`
Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).
Find the area of the parallelogram whose adjacent sides are determined by the vector `veca = hati - hatj + 3hatk` and `vecb = 2hati - 7hatj + hatk`.
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 ______.
If θ is the angle between two vectors `hati - 2hatj + 3hatk and 3hati - 2hatj + hatk` find `sin theta`
Find a unit vector perpendicular to both the vectors \[4 \hat{ i } - \hat{ j } + 3 \hat{ k } \text{ and } - 2 \hat{ i } + \hat{ j } - 2 \hat{ k } .\]
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 angle between two vectors \[\vec{a} \text{ and } \vec{b}\] , if \[\left| \vec{a} \times \vec{b} \right| = \vec{a} \cdot \vec{b} .\]
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.
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 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 \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} .\]
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.
For any two vectors \[\vec{a} \text{ and } \vec{b}\] , prove that \[\left| \vec{a} \times \vec{b} \right|^2 = \begin{vmatrix}\vec{a} . \vec{a} & & \vec{a} . \vec{b} \\ \vec{b} . \vec{a} & & \vec{b} . \vec{b}\end{vmatrix}\]
Using vectors find the area of the triangle with vertices, A (2, 3, 5), B (3, 5, 8) and C (2, 7, 8).
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) .
For any two vectors \[\vec{a}\] and \[\vec{b}\] , find \[\vec{a} . \left( \vec{b} \times \vec{a} \right) .\]
If \[\vec{c}\] is a unit vector perpendicular to the vectors \[\vec{a} \text{ and } \vec{b} ,\] write another unit vector perpendicular to \[\vec{a} \text{ and } \vec{b} .\]
Vectors \[\vec{a} \text{ and } \vec{b}\] \[\left| \vec{a} \right| = \sqrt{3}, \left| \vec{b} \right| = \frac{2}{3}\text{ and } \left( \vec{a} \times \vec{b} \right)\] is a unit vector. Write the angle between \[\vec{a} \text{ and } \vec{b}\] .
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
If \[\left| \vec{a} \times \vec{b} \right| = 4, \left| \vec{a} \cdot \vec{b} \right| = 2, \text{ then } \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 =\]
(a) If `veca = hati - 2j + 3veck , vecb = 2hati + 3hatj - 5hatk,` prove that `veca and vecaxxvecb` are perpendicular.
Let `veca` and `vecb` be two unit vectors and θ is the angle between them, Then `veca + vecb` is a unit vector if-
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 ______.
