let a^,b^, and c^ be unit vectors and α,β, and γ the angles between the vectors a^,b^;b^,c^, and c^,a^ respectively. a^+b^+c^ is also a unit vector, them cos⁡α+cos⁡β+cos⁡γ is equal to

a^⋅b^=cos⁡α, b^,c^=cos⁡β, c^⋅a^=cos⁡γ Now |a^+b^+c^|=1∴(a^+b^+c^)2=1⇒|a^|2+|b^|2+|c^|2+2(a^⋅b^+b^,c^+c^⋅a^)=1⇒1+1+1+2(cos⁡α+cos⁡β+cos⁡γ)=1⇒cos⁡α+cos⁡β+cos⁡γ=−1

let a^,b^, and c^ be unit vectors and α,β, and γ the angles between the vectors a^,b^;b^,c^, and c^,a^ respectively. a^+b^+c^ is also a unit vector, them cos⁡α+cos⁡β+cos⁡γ is equal to

a^⋅b^=cos⁡α, b^,c^=cos⁡β, c^⋅a^=cos⁡γ Now |a^+b^+c^|=1∴(a^+b^+c^)2=1⇒|a^|2+|b^|2+|c^|2+2(a^⋅b^+b^,c^+c^⋅a^)=1⇒1+1+1+2(cos⁡α+cos⁡β+cos⁡γ)=1⇒cos⁡α+cos⁡β+cos⁡γ=−1

AI Mentor

Check Your IQ

Free Expert Demo

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

let $\hat{a}, \hat{b}, and \hat{c}$ be unit vectors and $α,β, and γ$ the angles between the vectors $\hat{a}, \hat{b}; \hat{b}, \hat{c}, and \hat{c}, \hat{a}$ respectively. $\hat{a} + \hat{b} + \hat{c}$ is also a unit vector, them $\cos α + \cos β + \cos γ$ is equal to

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

-1

-3

answer is A.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

$\begin{array}{l} \hat{a} \cdot \hat{b} = \cos α, \hat{b}, \hat{c} = \cos β, \hat{c} \cdot \hat{a} = \cos γ \\ Now | \hat{a} + \hat{b} + \hat{c} | = 1 ∴ (\hat{a} + \hat{b} + \hat{c})^{2} = 1 \\ \Rightarrow | \hat{a} |^{2} + | \hat{b} |^{2} + | \hat{c} |^{2} + 2 (\hat{a} \cdot \hat{b} + \hat{b}, \hat{c} + \hat{c} \cdot \hat{a}) = 1 \\ \Rightarrow 1 + 1 + 1 + 2 (\cos α + \cos β + \cos γ) = 1 \\ \Rightarrow \cos α + \cos β + \cos γ = - 1 \end{array}$