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The position vectors of the vertices A, ...

The position vectors of the vertices A, B and C of a triangle are three unit vectors `veca, vecb and vecc` respectively. A vector `vecd` is such that `vecd.hata =vecd. Hatb=vecd.hatc and vecd= lambda (hatb + hatc)` . Then triangle ABC is

A

acute angled

B

obtuse angled

C

right angled

D

none of these

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The correct Answer is:
a
`vecd .hata = vecd .hatb = vecd .hatc`
` Rightarrow lambda (hata.hatb + hata.hatc) = lambda ( 1 + hatb.hatc) = lambda ( 1 + hatb .hatc)`
`or 1 + hatb .hatc = hata.hatb +hata.hatc`
`or 1 - hata.hatb + hatc - hatc .hatc =0`
`or 1- hata .hatb + (hatb- hata).hatc=0`
`or (hata -hatc) . (hata -hatb) =0 " Hence " hata - hatc` is perpendicular to `( hata-hatb)` i.e., the triangle is right angled.
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CENGAGE-DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS -Exercise
  1. veca and vecb are two unit vectors that are mutually perpendicular. A...

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  2. Given that veca,vecb,vecp,vecq are four vectors such that veca + vecb...

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  3. The position vectors of the vertices A, B and C of a triangle are thre...

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  4. If a is real constant A ,Ba n dC are variable angles and sqrt(a^2-4)ta...

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  5. The vertex A of triangle ABC is on the line vecr=hati+hatj+lambda hatk...

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  6. A non-zero vecto veca is such tha its projections along vectors (hati ...

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  7. Position vector hat k is rotated about the origin by angle 135^0 i...

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  8. ln a quadrilateral ABCD, vec(AC) is the bisector of the (vec(AB) ^^ v...

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  9. In fig. 2.33 AB, DE and GF are parallel to each other and AD, BG and E...

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  10. Vectors hata in the plane of vecb = 2 hati +hatj and vecc = hati-hatj ...

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  11. Let A B C D be a tetrahedron such that the edges A B ,A Ca n dA D a...

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  12. Let vecf(t)=[t] hat i+(t-[t]) hat j+[t+1] hat k , w h e r e[dot] deno...

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  13. If veca is parallel to vecb xx vecc, then (veca xx vecb) .(veca xx vec...

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  14. The three vectors hat i+hat j,hat j+hat k, hat k+hat i taken two at a ...

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  15. If vecd=vecaxxvecb+vecbxxvecc+veccxxveca is a on zero vector and |(vec...

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  16. If |veca|=2 and |vecb|=3 and veca.vecb=0, " then " (vecaxx(vecaxx(veca...

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  17. If two diagonals of one of its faces are 6hati + 6 hatk and 4 hatj + ...

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  18. The volume of a tetrahedron fomed by the coterminus edges veca , vecb ...

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  19. If veca ,vecb and vecc are three mutually orthogonal unit vectors , th...

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  20. vector vecc are perpendicular to vectors veca= (2,-3,1) and vecb= (1,...

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