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If vecaxxvecb=veccxxvecd and vecaxxvecc=...

If `vecaxxvecb=veccxxvecd and vecaxxvecc=vecbxxvecd` show that `(veca-vecd)` is parallel to `(vecb-vecc)`.

Text Solution

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Given,
`vecAxxvecB=vecCxxvecD`
`vecAxxvecC =vecBxxvecD`
Subtracting (2)from (1), we get ,
`vecAxxvecB-vecAxxvecC=vecCxxvecD -vecBxxvecD`
`:. vecAxx(vecB-vecC)`
`=(vecC -vecB)xxvecD[:.vecPxx(vecQ+vecR)=vecPxxvecQ+vecPxxvecR]`
`:. vecAxx(vecB-vecC)=-(vecB-vecC)xxvecD` `:. vecAxx(vecB-vecC)+(vecB-vecC)xxvecD=0 ` `:.vecAxx(vecB-vecC)-vecDxx(vecB-vecC)=0`
`[:.PxxvecQ=-vecQxxvecP]`
`:.(vecA-vecD)xx(vecB-vecC)=0`
Now,`|vecA|!=|vecD|"and"|vecB|!=|vecC|`
`:. (vecA-vecD)!=0"and" (vecB-vecC)!=0`
`(vecA-vecD)"and"(vecB-vecC)` are at an angle `0^(@)`or `(vecA-vecD)"and"(vecB-vecC)` are parallel.
`[:. vecPxxvecQ=PQsin theta "and"P!=0!=Q."hence " sin theta=0]`
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