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A , B , C , D are any four points, prove...

`A , B , C , D` are any four points, prove that ` vec A B dot vec C D+ vec B Cdot vec A D+ vec C Adot vec B D=0.`

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`vec(AD)=vec(AB)+vec(BC)+vec(CD)=veca+vecb+vecc`
`vec(AC)=vec(AB)+vec(BC)=veca+vecborvec(CA)=-(veca+vecb)`
`vec(BD)=vec(BC)=vec(CD)=vecb+vecc`
`vec(AB).vec(CD)+vec(BC).vec(AD)+vec(CA).vec(BD)`
`veca.vecc+vecb.(veca+vecb+vecc)+(=-veca-vecb).(vecb+vecc)`
`= veca.vecc+vecb.veca+vecb.vecb+vecb.vecc-veca.vecb-veca.vecc`
`-vecb.vecb-vecb.vecc=0`
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CENGAGE-DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS -Exercise 2.1
  1. Find |veca|and|vecb|, if (veca+vecb)*(veca-vecb)=8and|veca|=8|vecb|.

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  2. Show that |veca|vecb+|vecb|veca is perpendicular to |veca|vecb-|vecb|...

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  3. If the vertices A,B, C of a triangle ABC are (1,2,3),(-1, 0,0), (0, 1,...

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  4. If |a|=3,|b|=4a n d the angle between aa n db is 120^0 , then find the...

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  5. If vectors hati-2xhatj-3yhatk and hati+ 3 x hatj + 2yhatk are orthogon...

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  6. Let veca vecb and vecc be pairwise mutually perpendicular vectors, suc...

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  7. If veca +vecb +vecc=0, |veca|=3,|vecb|=5, |vecc|=7 , then find the ang...

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  8. If the angle between unit vectors veca and vecb is 60^(@) . Then find ...

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  9. Let vecu= hati+hatj, vecv = hati -hatj and vecw = hati+2hatj+3hatk. I...

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  10. A , B , C , D are any four points, prove that vec A B dot vec C D+ ...

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  11. P(1,0,-1), Q(2,0,-3),R(-1,2,0)a n dS(,-2,-1), then find the projection...

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  12. If the vectors 3vecP+vecq, 5vecP - 3vecq and 2 vecp+ vecq, 4 vecp - 2v...

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  13. Let vecA and vecB be two non-parallel unit vectors in a plane. If (alp...

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  14. If veca,vecb,vecc are three vectors such that veca+2vecb+vecc=vec0 and...

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  15. If veca and vecb are unit vectors, then find the greatest value of |ve...

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  16. Constant forces P1= hat i+ hat j+ hat k ,P2= hat i+2 hat j- hat ka n d...

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  17. If |veca|=5 , |veca-vecb| = 8 and |veca +vecb|=10 then find |vecb|

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  18. If A ,B ,C ,D are four distinct point in space such that A B is not...

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