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

Options

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

Answer

D

Explanation

Given that `veca =hati +hatj +hatk , vecb = 4hati +3hatj +4hatk and vecc =hati +alpha hatj + beta hatk` are linearly dependent,
`" "|{:(1,,1,,1),(4,,3,,4),(1,,alpha,,beta):}|=0`
or `1-beta =0`
or `beta =1 `
Also given that `|vecc|= sqrt3 rArr 1 + alpha^(2) + beta^(2) =3`
Subsituting the value of `beta`, we get
`alpha^(2) =1 or alpha = pm 1`