The number of integral values of p for which `(p+1) hati-3hatj+phatk, phati + (p+1)hatj-3hatk` and `-3hati+phatj+(p+1)hatk` are linearly dependent vectors is q

Options

A. 0
B. 1
C. 2
D. 3

Answer

B

Explanation

The vectors are linearly dependent
`rArr |{:(p+1,-3,p),(p,p+1,-3),(-3,p,p+1):}|=0`
`rArr (2p-2)|{:(1,-3,p),(1,p+1,-3),(1,p,p+1):}|=0`
`rArr 2(p-1)|{:(1,-3,p),(0,p+4,-3-p),(0,p+3,1):}|=0`
`rArr 2(p-1)(p+4)+(p+3)^(2)=0`
`rArr (p-1)(p^(2)+7p+13)=0`
Roots of `p^(2)+7p+13=0` are (imaginary)
`therefore p=1`
Only integral value of p is 1.