Advertisements
Advertisements
प्रश्न
If `bar"c" = 3bar"a" - 2bar"b"`, then prove that `[bar"a" bar"b" bar"c"] = 0`.
उत्तर
We use the results: `bar"b" xx bar"b" = bar"0"` and if in a scalar triple product, two vectors are equal, then the scalar triple product is zero.
`[bar"a" bar"b" bar"c"] = bar"a".(bar"b" xx bar"c")`
`= bar"a".[bar"b" xx (3bar"a" - 2bar"b")]`
`= bar"a".(3bar"b" xx bar"a" - 2bar"b" xx bar"b")`
`= bar"a". (3bar"b" xx bar"a" - bar"0")`
`= 3bar"a".(bar"b" xx bar"a")`
= 3 × 0
= 0
Alternative Method:
`bar"c" = 3bar"a" - 2bar"b"`
∴ `bar"c"` is a linear combination of `bar"a" "and" bar"b"`.
∴ `bar"a" , bar"b" , bar"c"` are coplanar.
∴ `[bar"a" bar"b" bar"c"] = 0`
APPEARS IN
संबंधित प्रश्न
Prove that `[bar"a" bar"b" + bar"c" bar"a" + bar"b" + bar"c"] = 0`
If `bar "a" = hat"i" + 2hat"j" + 3hat"k" , bar"b" = 3hat"i" + 2hat"j"` and `bar"c" = 2hat"i" + hat"j" + 3hat"k"`, then verify that `bar"a" xx (bar"b" xx bar"c") = (bar"a".bar"c")bar"b" - (bar"a".bar"b")bar"c"`
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.
Show that the points A(2, –1, 0) B(–3, 0, 4), C(–1, –1, 4) and D(0, – 5, 2) are non coplanar
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" = 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")`
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"`
Prove that `[vec"a" - vec"b", vec"b" - vec"c", vec"c" - vec"a"]` = 0
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")`
If `vec"a", vec"b", vec"c", vec"d"` are coplanar vectors, show that `(vec"a" xx vec"b") xx (vec"c" xx vec"d") = vec0`
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"`
`bar"a" xx (bar"b" xx bar"c") + bar"b" xx (bar"c" xx bar"a") + bar"c" xx (bar"a" xx bar"b")` = ?
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 `bar c = 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 `bar"c" = 3bar"a"-2bar"b"` and `[bar"a" bar"b" +bar"c" bar"a" +bar"b" +bar"c"]` = 0 then prove that `[bar"a" bar"b" bar"c"]` = 0
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` 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.
If, `barc = 3bara -2barb, "then prove that" [bara barb barc] = 0`
