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The position vectors of the vertices `A ,Ba n dC` of a triangle are three unit vectors ` vec a , vec b ,a n d vec c ,` respectively. A vector ` vec d` is such that ` vec ddot vec a= vec ddot vec ba n d vec d=lambda( vec b+ vec c)dot` Then triangle `A B C` is a. acute angled b. obtuse angled c. right angled d. none of these

A

acute angled

B

obtuse angled

C

right angled

D

none of these

Text Solution

Verified by Experts

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
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  3. The position vectors of the vertices A ,Ba n dC of a triangle are t...

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

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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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  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 the two diagonals of one its faces are 6 hat i+6 hat ka n d4 hat ...

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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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