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The number of integral values of p for w...

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

A

0

B

1

C

2

D

3

Text Solution

Verified by Experts

The correct Answer is:
B
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.
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Knowledge Check

  • The value of p so that vectors (2hati - hatj +),( hati +2hatj-3hatk) and (3hati + phatj + 5hatk) are coplaner should be:

    A
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    B
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    C
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    A
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    B
    `alpha=1,beta=+-1`
    C
    `alpha=-1,beta=+-1`
    D
    `alpha=+-1,beta=1`

  • If veca=hati+hatj+hatk, vecb=4hati+3hatj+4hatk and vecc=hati+alphahatj+betahatk are linearly dependent vectors and |vecc|=sqrt(3) then

    A
    `alpha=1, beta=-1`
    B
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    C
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    D
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