Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Prove that `[bar"a" bar"b" + bar"c" 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.
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" = 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 `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 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 ______
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 `bar c = 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 \[\overline{\mathrm{a}}=4\hat{\mathrm{i}}+3\hat{\mathrm{j}}+\hat{\mathrm{k}},\overline{\mathrm{b}}=\hat{\mathrm{i}}-2\hat{\mathrm{j}}+2\hat{\mathrm{k}}\] then \[\mathbf{\overline{a}}\times\left(\mathbf{\overline{a}}\times\left(\mathbf{\overline{a}}\times\mathbf{\overline{b}}\right)\right)=\]
