Advertisements
Advertisements
प्रश्न
If \[\vec{a}\] is any vector, then \[\left( \vec{a} \times \hat{ i } \right)^2 + \left( \vec{a} \times \hat{ j } \right)^2 + \left( \vec{a} \times \hat{ k } \right)^2 =\]
विकल्प
- \[\vec{a}^2\]
\[2 \vec{a}^2\]
- \[3 \vec{a}^2\]
\[4 \vec{a}^2\]
Advertisements
उत्तर
\[2 \vec{a}^2\]
\[\text{ Let } \vec{a} = a_1 \hat{ i } + a_2 \hat{ j } + a_3 \hat{ k } \]
\[ \vec{a} \times \hat{ i } = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ 1 & 0 & 0\end{vmatrix}\]
\[ = a_3 \hat{ j } - a_2 \hat{ k } \]
\[ \Rightarrow \left( \vec{a} \times \hat{ i } \right)^2 = \left( a_3 \hat{ j } - a_2 \hat{ k } \right)^2 \]
\[ = {a_3}^2 \left| \hat{ j } \right|^2 + {a_2}^2 \left| \hat{ k } \right|^2 - 2 a_3 a_2 \left( \hat{ j } . \hat{ k } \right)\]
\[ = {a_3}^2 + {a_2}^2 (\because \hat{ j } . \hat{ k } =0) . . . (1)\]
\[ \therefore \vec{a} \times \hat{ j } = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ 0 & 1 & 0\end{vmatrix}\]
\[ = - a_3 \hat{ i } + a_1 \hat{ k } \]
\[ \Rightarrow \left( \vec{a} \times \hat{ j } \right)^2 = \left( - a_3 \hat{ i } + a_1 \hat{ k } \right)^2 \]
\[ = {a_3}^2 \left| \hat{ i } \right|^2 + {a_1}^2 \left| \hat{ k } \right|^2 - 2 a_3 a_2 \left( \hat{ i} . \hat{ k } \right)\]
\[ = {a_3}^2 + {a_1}^2 (\because \hat{ i } .\hat{ k } =0) . . . (2)\]
\[ \therefore \vec{a} \times \hat{ k } = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ a_1 & a_2 & a_3 \\ 0 & 0 & 1\end{vmatrix}\]
\[ = a_2 \hat{ i } - a_1 \hat{ j } \]
\[ \Rightarrow \left( \vec{a} \times k \right)^2 = \left( a_2 \hat{ i} - a_1 \hat{ j} \right)^2 \]
\[ = {a_2}^2 \left| \hat{ i } \right|^2 + {a_1}^2 \left| j \right|^2 + 2 a_1 a_2 \left( \hat{ i } . \hat{ j },\right)\]
\[ = {a_2}^2 + {a_1}^2 (\because \hat{ i } . \hat{ j } =0) . . . (3)\]
\[\text{ Adding (1), (2) and (3), we get } \]
\[ \left( \vec{a} \times \hat{ i } \right)^2 + \left( \vec{a} \times \hat{ j } \right)^2 + \left( \vec{a} \times k \right)^2 = {a_3}^2 + {a_2}^2 + {a_3}^2 + {a_1}^2 + {a_2}^2 + {a_1}^2 \]
\[ = 2 \left( {a_1}^2 + {a_2}^2 + {a_3}^2 \right)\]
\[ = 2 \vec{a}^2 (\because\left| \vec{a} \right|=\sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2})\]
APPEARS IN
संबंधित प्रश्न
Find λ and μ if `(2hati + 6hatj + 27hatk) xx (hati + lambdahatj + muhatk) = vec0`.
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.`
If either `veca = vec0` or `vecb = vec0`, then `veca xxvecb = vec0`. Is the converse true? Justify your answer with an example.
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 ______.
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 \[\vec{a} = 2 \hat{ i } + \hat{ k } , \vec{b} = \hat { i } + \hat{ j } + \hat{ k } ,\] find the magnitude of \[\vec{a} \times \vec{b} .\]
Find 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 \[2 \hat{ i } + 3 \hat{ j } + 6 \hat{ k } \text{ and } 3 \hat{ i } - 6 \hat{ j } + 2 \hat{ k } \]
What inference can you draw if \[\vec{a} \times \vec{b} = \vec{0} \text{ and } \vec{a} \cdot \vec{b} = 0 .\]
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 \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} .\]
Find a unit vector perpendicular to each of the vectors \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} , \text{ where } \vec{a} = 3 \hat{ i } + 2 \hat{ j } + 2 \hat{ k } \text{ and } \vec{b} = \hat{ i } + 2 \hat{ j } - 2 \hat{ k } .\]
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 \[\left( \vec{b} + \vec{c} \right)\] .
If either \[\vec{a} = \vec{0} \text{ or } \vec{b} = \vec{0} , \text{ then } \vec{a} \times \vec{b} = \vec{0} .\] Is the converse true? Justify your answer with an example.
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 \[\vec{a} \times \left( \vec{b} + \vec{c} \right) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c} .\]
Using vectors, find the area of the triangle with vertice A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5) .
Define vector product of two vectors.
Write the value \[\left( \hat{ i } \times \hat{ j } \right) \cdot \hat{ k } + \hat{ i } \cdot \hat{ j } .\]
Write the value of \[\hat{ i } × \left( \hat{ j } + \hat{ k } \right) + \hat{ j } × \left( \hat{ k } + \hat{ i } \right) + \hat{ k } × \left( \hat{ i } + \hat{ j } \right) .\]
Write the expression for the area of the parallelogram having \[\vec{a} \text{ and } \vec{b}\] as its diagonals.
For any two vectors \[\vec{a}\] and \[\vec{b}\] , find \[\vec{a} . \left( \vec{b} \times \vec{a} \right) .\]
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 \[\vec{a}\] is a unit vector such that \[\vec{a} \times \hat{ i } = \hat{ j } , \text{ find } \vec{a} . \hat{ i } \] .
Find λ, if \[\left( 2 \hat{ i } + 6 \hat{ j } + 14 \hat{ k } \right) \times \left( \hat{ i } - \lambda \hat{ j } + 7 \hat{ k } \right) = \vec{0} .\]
If \[\hat{ i } , \hat{ j } , \hat{ k } \] are unit vectors, then
(a) If `veca = hati - 2j + 3veck , vecb = 2hati + 3hatj - 5hatk,` prove that `veca and vecaxxvecb` are perpendicular.
The number of vectors of unit length perpendicular to the vectors `vec"a" = 2hat"i" + hat"j" + 2hat"k"` and `vec"b" = hat"j" + hat"k"` is ______.
Find the area of the triangle with vertices A(1, l, 2), (2, 3, 5) and (1, 5, 5).
Let `veca` and `vecb` be two unit vectors and θ is the angle between them, Then `veca + vecb` is a unit vector if-
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 = 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 ______.
Find the area of the parallelogram whose diagonals are `hati - 3hatj + hatk` and `hati + hatj + hatk`.
