Advertisements
Advertisements
Question
Prove that `[vec"a" - vec"b", vec"b" - vec"c", vec"c" - vec"a"]` = 0
Advertisements
Solution
`[vec"a" - vec"b", vec"b" - vec"c", vec"c" - vec"a"] = (vec"a" - vec"b")*[(vec"b" - vec"c") xx (vec"c" - vec"a")]`
= `(vec"a" - vec"b")*[vec"b" xx vec"c" - vec"b" xx vec"a" - vec"c" xx vec"c" + vec"c" xx vec"a"]`
= `vec"a"*(vec"b" xx vec"c") - vec"a"*(vec"b" xx vec"a") - vec"a"*(vec"c" xx vec"c") + vec"a"*(vec"c" xx vec"a") - vec"b"*(vec"b" xx vec"c") + vec"b"*(vec"b" xx vec"a") + vec"b"*(vec"c" xx vec"c") - vec"b"*(vec"c" xx vec"a")`
= `vec"a"*(vec"b" xx vec"c") - 0 - 0 + 0 - 0 + 0 + 0 - vec"b"*(vec"c" xx vec"a")`
= `[vec"a", vec"b", vec"c"] - [vec"a", vec"b", vec"c"]` = 0
Hence proved.
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"]`.
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 `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"`
If `vec"a" = 2hat"i" + 3hat"j" - hat"k", vec"b" = 3hat"i" + 5hat"j" + 2hat"k", vec"c" = - hat"i" - 2hat"j" + 3hat"k"`, verify that `(vec"a" xx vec"b") xx vec"c" = (vec"a"*vec"c")vec"b" - (vec"b" * vec"c")vec"a"`
If `vec"a" = 2hat"i" + 3hat"j" - hat"k", vec"b" = 3hat"i" + 5hat"j" + 2hat"k", vec"c" = - hat"i" - 2hat"j" + 3hat"k"`, verify that `vec"a" xx (vec"b" xx vec"c") = (vec"a"*vec"c")vec"b" - (vec"a"*vec"b")vec"c"`
`vec"a" = 2hat"i" + 3hat"j" - hat"k", vec"b" = -hat"i" + 2hat"j" - 4hat"k", vec"c" = hat"i" + hat"j" + hat"k"` then find the va;ue of `(vec"a" xx vec"b")*(vec"a" xx vec"c")`
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 ______.
Let `veca = hati + hatj + hatk` and `vecb = hatj - hatk`. If `vecc` is a vector such that `veca.vecc = vecb` and `veca.vecc` = 3, then `veca.(vecb.vecc)` is equal to ______.
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] = "then proved" [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`
