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प्रश्न
\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{b} + \vec{c} \right) \times \left( \vec{a} + \vec{b} + \vec{c} \right) =\]
विकल्प
0
\[\left[ \vec{a} \vec{b} \vec{c} \right]\]
\[2\left[ \vec{a} \vec{b} \vec{c} \right]\]
\[\left[ \vec{a} \vec{b} \vec{c} \right]\]
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उत्तर
\[ \left[ \vec{a} \vec{b} \vec{c} \right] \]
We have
\[\left( \vec{a} + \vec{b} \right) . \left( \vec{b} + \vec{c} \right) \times \left( \vec{a} + \vec{b} + \vec{c} \right)\]
\[ = \left( \vec{a} + \vec{b} \right) . \left[ \left( \vec{b} + \vec{c} \right) \times \vec{a} + \left( \vec{b} + \vec{c} \right) \times \vec{b} + \left( \vec{b} + \vec{c} \right) \times \vec{c} \right]\]
\[ = \left( \vec{a} + \vec{b} \right) . \left( \vec{b} \times \vec{a} + \vec{c} \times \vec{a} + \vec{b} \times \vec{b} + \vec{c} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{c} \right)\]
\[ = \left( \vec{a} + \vec{b} \right) . \left( \vec{b} \times \vec{a} + \vec{c} \times \vec{a} + 0 + \vec{c} \times \vec{b} + \vec{b} \times \vec{c} + 0 \right) \]
\[ = \left( \vec{a} + \vec{b} \right) . \left( \vec{b} \times \vec{a} + \vec{c} \times \vec{a} - \vec{b} \times \vec{c} + \vec{b} \times \vec{c} \right)\]
\[ = \left( \vec{a} + \vec{b} \right) . \left( \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \right)\]
\[ = \vec{a} . \left( \vec{b} \times \vec{a} \right) + \vec{b} . \left( \vec{b} \times \vec{a} \right) + \vec{a} . \left( \vec{c} \times \vec{a} \right) + \vec{b} . \left( \vec{c} \times \vec{a} \right)\]
\[ = 0 + 0 + 0 + \left[ \vec{b} \vec{c} \vec{a} \right] \]
\[ = \left[ \vec{b} \vec{c} \vec{a} \right] \]
\[ = \left[ \vec{a} \vec{b} \vec{c} \right] \]
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संबंधित प्रश्न
If A, B, C, D are (1, 1, 1), (2, 1, 3), (3, 2, 2), (3, 3, 4) respectively, then find the volume of parallelopiped with AB, AC and AD as the concurrent edges.
Find the volume of the parallelopiped whose coterminus edges are given by vectors
`2hati+3hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk`
If `bara=3hati-hatj+4hatk, barb=2hati+3hatj-hatk, barc=-5hati+2hatj+3hatk` then `bara.(barbxxbarc)=`
(A) 100
(B) 101
(C) 110
(D) 109
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Prove that a necessary and sufficient condition for three vectors \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] to be coplanar is that there exist scalars l, m, n not all zero simultaneously such that \[l \vec{a} + m \vec{b} + n \vec{c} = \vec{0} .\]
Evaluate the following:
\[\left[\hat{i}\hat{j}\hat{k} \right] + \left[ \hat{j}\hat{k}\hat {i} \right] + \left[ \hat{k}\hat{i} \hat{j} \right]\]
Find \[\left[ \vec{a} \vec{b} \vec{c} \right]\] , when \[\vec{a} = 2 \hat{i} - 3 \hat{j} , \vec{b} = \hat{i} + \hat{j} - \hat{k} \text{ and } \vec{c} = 3 \hat{i} - \hat{k}\]
Find the volume of the parallelopiped whose coterminous edges are represented by the vector:
\[\vec{a} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k} , \vec{b} = \hat{i} + 2 \hat{j} - \hat{k} , \vec{c} = 3 \hat{i} - \hat{j} - 2 \hat{k}\]
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\[\vec{a} = 11 \hat{i} , \vec{b} = 2 \hat{j} , \vec{c} = 13 \hat{k}\]
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\[\vec{a} = - 4 \hat{i} - 6 \hat{j} - 2 \hat{k} , \vec{b} = -\hat{ i} + 4 \hat{j} + 3 \hat{k} , \vec{c} = - 8 \hat{i} - \hat{j} + 3 \hat{k}\]
Find the value of λ so that the following vector is coplanar:
\[\vec{a} = \hat{i} - \hat{j} + \hat{k} , \vec{b} = 2 \hat {i} + \hat {j} - \hat {k} , \vec{c} = \lambda\hat { i} - \hat {j} + \lambda \hat {k}\]
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\[\text {Let } \vec{a} = \hat {i} + \hat {j} + \hat {k} , \vec{b} = \hat {i} \text{and} \hat {c} = c_1 \hat{i} + c_2 \hat {j} + c_3 \hat {k} . \text {Then},\]
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If \[\left[ 3 \vec{a} + 7 \vec{b} \vec{c} \vec{d} \right] = \lambda\left[ \vec{a} \vec{c} \vec{d} \right] + \mu\left[ \vec{b} \vec{c} \vec{d} \right],\] then find the value of λ + μ.
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Determine where `bar"a"` and `bar"b"` are orthogonal, parallel or neithe:
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Ler `vec"a" = hat"i" + hat"j" + hat"k", vec"b" = hat"i"` and `vec"c" = "c"_1hat"i" + "c"_2hat"j" + "c"_3hat"k"`. If c1 = 1 and c2 = 2. find c3 such that `vec"a", vec"b"` and `vec"c"` are coplanar
The volume of tetrahedron whose vertices are A(3, 7, 4), B(5, -2, 3), C(-4, 5, 6), D(1, 2, 3) is ______.
If the scalar triple product of the vectors `-3hat"i" + 7hat"j" - 3hat"k", 3hat"i" - 7hat"j" + lambdahat"k" and 7hat"i" - 5hat"j" - 5hat"j"` is 272 then λ = ______.
If `veca, vecb, vecc` are three non-coplanar vectors, then the value of `(veca.(vecb xx vecc))/((vecc xx veca).vecb) + (vecb.(veca xx vecc))/(vecc.(veca xx vecb))` is ______.
Determine whether `bara and barb` are orthogonal, parallel or neither.
`bara = -3/5hati + 1/2hatj + 1/3hatk, barb = 5hati + 4hatj + 3hatk`
Determine whether `bb(bara and barb)` are orthogonal, parallel or neither.
`bar a = -3/5hati + 1/2hatj + 1/3hatk, barb = 5hati + 4hatj + 3hatk`
If `2hati + 3hatj, hati + hatj + hatk` and `λhati + 4hatj + 2hatk` taken in order are coterminous edges of a parallelopiped of volume 2 cu. units, then find the value of λ.
Determine whether `bara and barb` are orthogonal, parallel or neither.
`bara = -3/5hati + 1/2hatj +1/3 hatk, barb = 5hati + 4hatj +3hatk`
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Determine whether `bara and barb` are orthogonal, parallel or neither.
`bara = -3/5 hati + 1/2 hatj + 1/3 hatk, barb = 5 hati + 4 hatj + 3 hatk`
The magnitude of a vector which is orthogonal to the vector \[\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\] and is coplanar with the vectors \[\hat{\mathrm{i}}+\hat{\mathrm{j}}+2\hat{\mathrm{k}}\] and \[\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}}\] is
