Consider three vectors `A=hati+hatj-2hatk,B=hati-hatj+hatk` and `C=2hati-3hatj+4hatk`. A vector X of the form `alphaA+betaB`(`alpha` and `beta` are numbers) is perpendicular to C. The ratio of `alpha` and `beta` is

Options

A. `1 : 1`
B. `2 : 1`
C. `-1 : 1`
D. `3 : 1`

Answer

A

Explanation

The given vector, `X=alphaA+betaB=alpha(hati+hatj-2hatk)+beta(hati-hatj+hatk)`
`=(alpha+beta)hati+(alpha-beta)hatj+(-2alpha+beta)hatk`
and `C=2hati-3hatj+4hatk`
Now, `alphaA+betaB` is perpendicular to C when `(alphaA+betaB).C=0`
`implies(alpha+beta)(2)+(alpha-beta)(-3)+(-2alpha+beta)(4)=0`
`implies2alpha+2beta-3alpha+3beta-8alpha+4beta=0`
`implies" "-9alpha+9beta=0`
`implies" "-9alpha=-9beta`
`implies" "(alpha)/(beta)=1`
`:." "alpha:beta=1 : 1`