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प्रश्न
\[\left[ \vec{a} \vec{b} \vec{a} \times \vec{b} \right] + \left( \vec{a} . \vec{b} \right)^2 =\]
पर्याय
\[\left| \vec{a} \right|^2 \left| \vec{b} \right|^2\]
\[\left| \vec{a} + \vec{b} \right|^2\]
\[\left| \vec{a} \right|^2 + \left| \vec{b} \right|^2\]
\[2 \left| \vec{a} \right|^2 \left| \vec{b} \right|^2\]
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उत्तर
\[\left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \]
We have
\[\left[ \vec{a} \vec{b} \vec{a} \times \vec{b} \right] + \left( \vec{a} . \vec{b} \right)^2 \]
\[ = \left( \vec{a} \times \vec{b} \right) . \left( \vec{a} \times \vec{b} \right) + \left( \vec{a} . \vec{b} \right)^2 \]
\[ = \left| \left( \vec{a} \times \vec{b} \right) \right|^2 + \left( \vec{a} . \vec{b} \right)^2 \]
\[ = \left( \left| \vec{a} \right|\left| \vec{b} \right| \sin \theta \right)^2 + \left( \left| \vec{a} \right| \left| \vec{b} \right|^{} \cos \theta \right)^2 \]
\[ = \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \sin^2 \theta + \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \cos^2 \theta\]
\[ = \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \left( \sin^2 \theta + \cos^2 \theta \right)\]
\[ = \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 \]
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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
Let `vec"a", vec"b", vec"c"` be three non-zero vectors such that `vec"c"` is a unit vector perpendicular to both `vec"a"` and `vec"b"`. If the angle between `vec"a"` and `vec"b"` is `pi/6`, show that `[(vec"a", vec"b", vec"c")]^2 = 1/4|vec"a"|^2|vec"b"|^2`
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`bara=-3/5hati+1/2hatj+1/3hatk,barb=5hati+4hatj+3hatk`
Find the volume of a tetrahedron whose vertices are A(−1, 2, 3) B(3, −2, 1), C (2, 1, 3) and D(−1, −2, 4).
Find the volume of a tetrahedron whose vertices are A(−1, 2, 3) B(3, −2, 1), C(2, 1, 3) and D(−1, −2, 4).
