Advertisements
Advertisements
Question
For any two vectors \[\vec{a} \text{ and } \vec{b} , \text{ find } \left( \vec{a} \times \vec{b} \right) . \vec{b} .\]
Solution
\[\text{ Let } :\]
\[ \vec{a} = a_1 \hat{ i } + a_2 \hat{ j } + a_3 \hat{ k } \]
\[ \vec{b} = b_1 \hat{ i } + b_2 \hat{ j } + b_3 \hat{ k } \]
\[ \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\end{vmatrix}\]
\[ = \hat{ i } \left( a_2 b_3 - a_3 b_2 \right) - \hat{ j } \left( a_1 b_3 - a_3 b_1 \right) + \hat{ k } \left( a_1 b_2 - a_2 b_1 \right)\]
\[\left( \vec{a} \times \vec{b} \right) . \vec{b} \]
\[ = \left[ \hat{ i } \left( a_2 b_3 - a_3 b_2 \right) - \hat{ j } \left( a_1 b_3 - a_3 b_1 \right) + \hat{ k } \left( a_1 b_2 - a_2 b_1 \right) \right] . \left( b_1 \hat{ i } + b_2 \hat{ j } + b_3 \hat{ k } \right)\]
\[ = b_1 \left( a_2 b_3 - a_3 b_2 \right) - b_2 \left( a_1 b_3 - a_3 b_1 \right) + b_3 \left( a_1 b_2 - a_2 b_1 \right)\]
\[ = a_2 b_1 b_3 - a_3 b_1 b_2 - a_1 b_2 b_3 + a_3 b_1 b_2 + a_1 b_2 b_3 - a_2 b_1 b_3 \]
\[ = 0\]
APPEARS IN
RELATED QUESTIONS
Find `|veca × vecb|`, if `veca = hati - 7hatj + 7hatk` and `vecb = 3hati - 2hatj + 2hatk`.
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).
Area of a rectangle having vertices A, B, C, and D with position vectors `-hati + 1/2 hatj + 4hatk, hati + 1/2 hatj + 4hatk, and -hati - 1/2j + 4hatk,` respectively is ______.
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 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 angle between two vectors \[\vec{a} \text{ and } \vec{b}\] , if \[\left| \vec{a} \times \vec{b} \right| = \vec{a} \cdot \vec{b} .\]
What inference can you draw if \[\vec{a} \times \vec{b} = \vec{0} \text{ and } \vec{a} \cdot \vec{b} = 0 .\]
If \[\vec{a,} \vec{b,} \vec{c}\] are three unit vectors such that \[\vec{a} \times \vec{b} = \vec{c} , \vec{b} \times \vec{c} = \vec{a,} \vec{c} \times \vec{a} = \vec{b} .\] Show that \[\vec{a,} \vec{b,} \vec{c}\] form an orthonormal right handed triad of unit vectors.
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.
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 \[\vec{a} \text{ and } \vec{d}\] \[\text{ and } \vec{c} \cdot \vec{d} = 15 .\]
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.
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 \[\vec{a} \text{ and } \vec{b} .\]
For any three vectors \[\vec{a,} \vec{b} \text{ and } \vec{c}\] write the value of \[\vec{a} \times \left( \vec{b} + \vec{c} \right) + \vec{b} \times \left( \vec{c} + \vec{a} \right) + \vec{c} \times \left( \vec{a} + \vec{b} \right) .\]
Write a unit vector perpendicular to \[\hat{ i } + \hat{ j } \text{ and } \hat{ j } + \hat{ k } .\]
If \[\left| \vec{a} \times \vec{b} \right|^2 + \left( \vec{a} . \vec{b} \right)^2 = 144\] and \[\left| \vec{a} \right| = 4,\] find \[\left| \vec{b} \right|\] .
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 \[\vec{a} \text{ and } \vec{b} .\]
Find λ, if \[\left( 2 \hat{ i } + 6 \hat{ j } + 14 \hat{ k } \right) \times \left( \hat{ i } - \lambda \hat{ j } + 7 \hat{ k } \right) = \vec{0} .\]
Write the value of the area of the parallelogram determined by the vectors \[2 \hat{ i } \text{ and } 3 \hat{ j } .\]
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 \[\vec{a}\] is any vector, then \[\left( \vec{a} \times \hat{ i } \right)^2 + \left( \vec{a} \times \hat{ j } \right)^2 + \left( \vec{a} \times \hat{ k } \right)^2 =\]
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 θ is the angle between any two vectors `bara` and `barb` and `|bara · barb| = |bara xx barb|` then θ is equal to ______.
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.
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| = sqrt(3)` and `veca.vecb` = – 3, then angle between `veca` and `vecb` is ______.
Find the area of a parallelogram whose adjacent sides are determined by the vectors `veca = hati - hatj + 3hatk` and `vecb = 2hati - 7hatj + hatk`.
Find the area of the parallelogram whose diagonals are `hati - 3hatj + hatk` and `hati + hatj + hatk`.
