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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 Bdot vec C D+ vec B Cdot vec A D+ vec C Adot vec B D=4(Area \ of triangle ABC).`

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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 PUBLICATION-DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS -Exercise 2.1
  1. Find |veca|and |vecb|, if (veca+vecb).(veca-vecb) =8 and |veca|=8|vecb...

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  2. Show that | vec a| vec b+| vec b| vec a is a perpendicular to | vec a|...

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

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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, su...

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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 120^(@) . Then find...

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

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

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

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  12. If the vectors 3 vec p+ vec q ;5p-3 vec qa n d2 vec p+ vec q ;3 vec p-...

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  13. Let vec A and vec B be two non-parallel unit vectors in a plane. If...

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  14. Let vec a , vec ba n d vec c be unit vectors, such that vec a+ vec b...

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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...

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  17. Find |veca|and |vecb|, if (veca+vecb).(veca-vecb) =8 and |veca|=8|vecb...

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  18. If A ,B ,C ,D are four distinct point in space such that vec(A B) is...

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