Let a, b, c ∈ R be such that a2+b2+c2=1 If a cos θ=b cos (θ+(2 π/3))=c cos (θ+(4 π/3)), where θ=(π/9), then the angle between the vectors a hati+b hatj+c hatk and b hati+c hatj+a hatk is :

Question

Let a, b, c ∈ R be such that a2+b2+c2=1 If a cos θ=b cos (θ+(2 π/3))=c cos (θ+(4 π/3)), where θ=(π/9), then the angle between the vectors a hati+b hatj+c hatk and b hati+c hatj+a hatk is : (A) (π/2) (B) 0 (C) (π/9) (D) (2 π/3)

Tardigrade Admin · Accepted Answer

cos φ=( overline p ⋅ overline q /| overline p || overline q |)=( ab + bc + ca / a 2+ b 2+ c 2)=( Sigma ab /1) =abc ((1/ a )+(1/ b )+(1/ c )) =(abc/λ)( cos θ+ cos (θ+(2 π/3))+ cos (θ+(4 π/3))) =(abc/λ)( cos +2 cos (θ+π) cos (π/3)) =(a b c/λ)( cos θ- cos θ)=0 φ=(π/2)

Q. Let $a, b, c \in R$ be such that $a^{2} + b^{2} + c^{2} = 1$ If a $cos θ = b cos (θ + \frac{2 π}{3}) = c cos (θ + \frac{4 π}{3})$ , where $θ = \frac{π}{9}$ , then the angle between the vectors $a \hat{i} + b \hat{j} + c \hat{k}$ and $b \hat{i} + c \hat{j} + a \hat{k}$ is :

$\frac{π}{2}$

0

$\frac{π}{9}$

$\frac{2 π}{3}$

Q. Let a,b,c∈R be such that a2+b2+c2=1 If a cosθ=bcos(θ+32π​)=ccos(θ+34π​), where θ=9π​, then the angle between the vectors ai^+bj^​+ck^ and bi^+cj^​+ak^ is :

2π​

0

9π​

32π​

cosϕ=∣p​∣∣q​∣p​⋅q​​=a2+b2+c2ab+bc+ca​=1Σab​ =abc(a1​+b1​+c1​) =λabc​(cosθ+cos(θ+32π​)+cos(θ+34π​)) =λabc​(cos+2cos(θ+π)cos3π​) =λabc​(cosθ−cosθ)=0 ϕ=2π​

Q. Let $a, b, c \in R$ be such that $a^{2} + b^{2} + c^{2} = 1$ If a $cos θ = b cos (θ + \frac{2 π}{3}) = c cos (θ + \frac{4 π}{3})$ , where $θ = \frac{π}{9}$ , then the angle between the vectors $a \hat{i} + b \hat{j} + c \hat{k}$ and $b \hat{i} + c \hat{j} + a \hat{k}$ is :

$\frac{π}{2}$

$\frac{π}{9}$

$\frac{2 π}{3}$