Advertisements
Advertisements
प्रश्न
Prove that, for any three vector `veca,vecb,vecc [vec a+vec b,vec b+vec c,vecc+veca]=2[veca vecb vecc]`
उत्तर
We have :
`[vec a+vec b,vec b+vec c,vecc+veca]`
`=(veca+vecb).[(vecb+vecc)xx(vecc+veca)]`
`=(veca+vecb).[vecbxxvecc+vecbxxveca+vecc xx vecc+vec c xx vec a] (By distributive law)`
`=(veca+vecb).[vecbxxvecc+vecbxxveca+vec c xx vec a] (∵vecc xx vecc=0)`
`=veca.(vecbxxvecc)+vecb(vecb xx vecc)+vec a(vecb xx veca+vecb.(vecb xx veca))+veca.(vecc xxveca)+vecb.(vecc xx veca)`
`=[veca,vecb,vecc]+[vecb,vecb,vecc]+[veca,vecb,veca]+[vecb,vecb,veca]+[veca,vecc,veca]+[vecb,vecc,veca]`
`=[veca,vecb,vecc]+[vecb,vecc,veca] " (∵scalar triple product with two equal vectors is 0) "`
`=[veca,vecb,vecc]+[veca,vecb,vecc] (because [vecb,vecc,veca]=[veca,vecb,vecc])`
`=2[veca,vecb,vecc]`
Hence,
`[vec a+vec b,vec b+vec c,vecc+veca]=2[veca vecb vecc]`
संबंधित प्रश्न
If `bara=3hati-hatj+4hatk, barb=2hati+3hatj-hatk, barc=-5hati+2hatj+3hatk` then `bara.(barbxxbarc)=`
(A) 100
(B) 101
(C) 110
(D) 109
Find the volume of the parallelopiped whose coterminus edges are given by vectors `2hati+5hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk`
Find the volume of the parallelopiped whose coterminous edges are represented by the vector:
\[\vec{a} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k} , \vec{b} = \hat{i} + 2 \hat{j} - \hat{k} , \vec{c} = 3 \hat{i} - \hat{j} - 2 \hat{k}\]
Show of the following triad of vector is coplanar:
\[\vec{a} = - 4 \hat{i} - 6 \hat{j} - 2 \hat{k} , \vec{b} = -\hat{ i} + 4 \hat{j} + 3 \hat{k} , \vec{c} = - 8 \hat{i} - \hat{j} + 3 \hat{k}\]
Show of the following triad of vector is coplanar:
\[\hat{a} = \hat{i} - 2 \hat {j} + 3 \hat {k} , \hat {b} = - 2 \hat {i} + 3 \hat {j} - 4 \hat { k}, \hat {c} = \hat { i} - 3 \hat { j} + 5 \hat { k }\]
Find the value of λ so that the following vector is coplanar:
\[\vec{a} = \hat {i} + 3 \hat {j} , \vec{b} = 5 \hat {k} , \vec{c} = \lambda \hat {i} - \hat {j}\]
Write the value of \[\left[ \hat {i} + \hat {j} \ \hat {j} + \hat {k} \ \hat {k} + \hat {i} \right] .\]
If \[\vec{a,} \vec{b,} \vec{c}\] are non-coplanar vectors, then find the value of \[\frac{\vec{a} \cdot \left( \vec{b} \times \vec{c} \right)}{\left( \vec{c} \times \vec{a} \right) \cdot \vec{b}} + \frac{\vec{b} \cdot \left( \vec{a} \times \vec{c} \right)}{\vec{c} \cdot \left( \vec{a} \times \vec{b} \right)}\].
If \[\vec{a}\] lies in the plane of vectors \[\vec{b} \text { and } \vec{c}\], then which of the following is correct?
If \[\vec{a,} \vec{b,} \vec{c}\] are three non-coplanar mutually perpendicular unit vectors, then \[\left[ \vec{a} \vec{b} \vec{c} \right],\] is
\[\left[ \vec{a} \vec{b} \vec{a} \times \vec{b} \right] + \left( \vec{a} . \vec{b} \right)^2 =\]
If a vector has direction angles 45° and 60°, find the third direction angle.
Find `bar"a".(bar"b" xx bar"c")` if `bar"a" = 3hat"i" - hat"j" + 4hat"k" , bar"b" = 2hat"i" + 3hat"j" - hat"k"` and `bar"c" = - 5hat"i" + 2hat"j" + 3hat"k"`
If the vectors `- 3hat"i" + 4hat"j" - 2hat"k" , hat"i" + 2hat"k"` and `hat"i" - "p"hat"j"` are coplanar, then find the value of p.
If `bar"u" = hat"i" - 2hat"j" + hat"k" , bar"v" = 3hat"i" + hat"k"` and `bar"w" = hat"j" - hat"k"` are given vectors, then find `(bar"u" + bar"w").[(bar"u" xx bar"v") xx (bar"v" xx bar"w")]`
Using properties of scalar triple product, prove that `[(bar"a" + bar"b", bar"b" + bar"c", bar"c" + bar"a")] = 2[(bar"a", bar"b", bar"c")]`.
Determine whether the three vectors `2hat"i" + 3hat"j" + hat"k", hat"i" - 2hat"j" + 2hat"k"` and `3hat"i" + hat"j" + 3hat"k"` are coplanar
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
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 the volume of tetrahedron whose vertices are A(0, 1, 2), B(2, -3, 0), C(1, 0, 2) and D(-2,-3,lambda) is `7/3` cu.units, then the value of λ is ______.
If the points having the position vectors `2hat"i" + hat"j" - hat"k", -hat"j", 4hat"i" + 4hat"k"` and `lambdahat"i" + hat"k"` lie on the same plane, then λ is equal to ______.
If the direction cosines of a line are `(1/c, 1/c, 1/c)` then ______.
Determine whether `bara and barb` are orthogonal, parallel or neither.
`bara = - 3/5 hati+ 1/2 hatj + 1/3 hatk , barb= 5hati + 4hatj + 3hatk`
Determine whether `bara` and `barb` are orthogonal, parallel or neither.
`bara = - 3/5 hati + 1/2 hatj + 1/3 hatk, 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.
Find the volume of the parallelopiped whose vertices are A (3, 2, −1), B (−2, 2, −3) C (3, 5, −2) and D (−2, 5, 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).
