Advertisements
Advertisements
Question
For any vector `vec"a"`, prove that `hat"i" xx (vec"a" xx hat"i") + hat"j" xx (vec"a" xx hat"j") + hat"k" xx (vec"a" xx hat"k") = 2vec"a"`
Advertisements
Solution
Let `vec"a" = "a"_1hat"i" + "a"_2hat"j" + "a"_3hat"k"`
`vec"a" xx hat"i" = |(hat"i", hat"j", hat"k"),("a"_1, "a"_2, "a"_3),(1, 0, 0)|`
= `hat"i"(0) - hat"j"(- "a"_3) + hat"k"(0 - "a"_2)`
`hat"i" xx (vec"a" xx hat"i") = |(hat"i", hat"j", hat"k"),(1, 0, 0),(0, "a"_3, - "a"_2)|`
= `hat"i"(0) - hat"j"(- "a"_2) + hat"k"("a"_3)`
= `"a"_2hat"j" + "a"_3hat"k"`
Similarly `hat"j" xx (vec"a" xx hat"j") = "a"_1hat"i" + "a"_3hat"k"`
`hat"k" xx (vec"a" xx hat"k") - "a"_1hat"i" + "a"_2hat"j"`
`hat"i" xx (vec"a" xx hat"i") + hat"j" xx (vec"a" xx hat"j") + hat"k" xx (vec"a" xx hat"k")`
= `2"a"_1hat"i" + 2"a"_2hat"j" + 2"a"_3hat"k"`
= `2("a"_1hat"i" + "a"_2hat"j" + "a"_3hat"k")`
= `2hat"a"`
APPEARS IN
RELATED QUESTIONS
Prove that `[bar"a" bar"b" + bar"c" bar"a" + bar"b" + bar"c"] = 0`
Prove that `(bar"a" + 2bar"b" - bar"c"). [(bar"a" - bar"b") xx (bar"a" - bar"b" - bar"c")] = 3 [bar"a" bar"b" bar"c"]`.
If `bar"c" = 3bar"a" - 2bar"b"`, then prove that `[bar"a" bar"b" bar"c"] = 0`.
Show that `bar"a" xx (bar"b" xx bar"c") + bar"b" xx (bar"c" xx bar"a") + bar"c" xx (bar"a" xx bar"b") = bar"0"`
If `bara = hati - 2hatj`, `barb = hati + 2hatj, barc = 2hati + hatj - 2hatk`, then find (i) `bara xx (barb xx barc)` (ii) `(bara xx barb) xx barc`. Are the results same? Justify.
If `vec"a" = hat"i" - 2hat"j" + 3hat"k", vec"b" = 2hat"i" + hat"j" - 2hat"k", vec"c" = 3hat"i" + 2hat"j" + hat"k"`, find `vec"a" xx (vec"b" xx vec"c")`
Prove that `[vec"a" - vec"b", vec"b" - vec"c", vec"c" - vec"a"]` = 0
If `hat"a", hat"b", hat"c"` are three unit vectors such that `hat"b"` and `hat"c"` are non-parallel and `hat"a" xx (hat"b" xx hat"c") = 1/2 hat"b"`, find the angle between `hat"a"` and `hat"c"`
If a, b, care non-coplanar vectors and p = `("b" xx "c")/(["abc"]), "q" = ("c" xx "a")/(["abc"]), "r" = ("a" xx "b")/(["abc"])`, then a · p + b · q + c · r = ?
Let A(4, 7, 8), B(2, 3, 4) and C(2, 5, 7) be the vertices of a triangle ABC. The length of the internal bisector of angle A is ______
If `bar"a" = 3hat"i" - 2hat"j" + 7hat"k", bar"b" = 5hat"i" + hat"j" - 2hat"k"` and `bar "c" = hat"i" + hat"j" - hat"k"`, then `[bar"a" bar"b" bar"c"]` = ______.
Let three vectors `veca, vecb` and `vecc` be such that `vecc` is coplanar with `veca` and `vecb, vecc,` = 7 and `vecb` is perpendicular to `vecc` where `veca = -hati + hatj + hatk` and `vecb = 2hati + hatk`, then the value of `2|veca + vecb + vecc|^2` is ______.
If `veca = hati + 2hatj + 3hatk, vecb = 2hati + 3hatj + hatk, vecc = 3hati + hatj + 2hatk` and `αveca + βvecb + γvecc = -3(hati - hatk)`, then the ordered triplet (α, β, γ) is ______.
`"If" barc=3bara-2barb "and" [bara barb+barc bara+barb+barc]= 0 "then prove that" [bara barb barc]=0 `
If `overlinec = 3overlinea - 2overlineb` and `[overlinea overlineb + overlinec overlinea + overlineb + overlinec]` = 0 then prove that `[overlinea overlineb overlinec]` = 0
Show that the volume of the parallelopiped whose coterminus edges are `bara barb barc` is `[(bara, barb, barc)].`
If `barc = 3bara - 2barb and [bara barb+barc bara+barb+barc] = 0` then prove that `[bara barb barc] = 0`
If `barc = 3bara - 2barb and [bara barb+barc bara + barb + barc] = 0` then prove that `[bara barb barc] = 0`
If `barc = 3bara - 2barb`, then prove that `[bara barb barc]` = 0.
