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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
\[\left[ \vec{a} \vec{b} \vec{c} \right]\]
\[2\left[ \vec{a} \vec{b} \vec{c} \right]\]
\[\left[ \vec{a} \vec{b} \vec{c} \right]\]
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
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if `bara = 3hati - 2hatj+7hatk`, `barb = 5hati + hatj -2hatk`and `barc = hati + hatj - hatk` then find `bara.(barbxxbarc)`
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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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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
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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 `\bb(bara and barb)` are orthogonal, parallel or neither.
`bara = -3/5 hati + 1/2 hatj + 1/3 hatk, barb = 5hati + 4hatj + 3hatk `
