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 d ⋅ vec a = vec d ⋅ vec b = vec d ⋅ vec c a 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

Options

A. acute angled
B. obtuse angled
C. right angled
D. none of these

Answer

a

Explanation

`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.