The angle between the vectors a and b having direction ratios (-p,1,-2) and (2,-p,-1) is `pi/3` then the value of p is

Options

A. `-3pm2sqrt2`
B. `-3pm2sqrt3`
C. `-3pm4sqrt3`
D. None of these

Answer

A

Explanation

Given `(a_1,b_1,c_1)=(-p,1,-2)`
`and (a_2,b_2,c_2)=(2,-p,-1)`
`therefore" "costheta=(a_1a_2+b_1b_2+c_1c_2)/(sqrt(a_1^2+b_1^2+c_1^2)sqrt(a_2^2+b_2^2+c_2^2))`
`cospi/3=1/2=(-pxx2+1xx(-p)+(-2)xx(-1))/(sqrt(p^2+1^2+2^2)sqrt(2^2+p^2+1^2))`
`rArr" "1/2=(-3p+2)/(sqrt(p^2+5)sqrt(p^2+5))`
`rArr" "1/2=(-3p+2)/(p^2+5)`
`rArrp^2+6p+1=0`
`therefore" "p=(-6pmsqrt((6)^2-4xx1))/(2xx1)=(-6pmsqrt(32))/2`
`=(-6pm4sqrt2)/2=-3pm2sqrt2`