Consider three vectors `A=hati +hatj-2hatk`,
`B=hati +hatj+2hatk and 2hati -3hatj +4hatk` A vector X of the from `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

(a) vector x of the perpendicular ro` ,C,i.,e X.C=0`
`x=alphaA +betaB`
`=alpha(hati+hatj-2hatk)+beta(hati-hatj+hatk)`
`X=hati(alpha+beta)+hatj(alpha-beta)+hatk(-2alpha+beta)` vectorer X is perpendiicular to C,i.e. X.C =-0
`[hati(alpha+beta)+(alpha-beta)+hatk(-2alpha+beta)].[2hati-3hatj+4hatk]=0`
`implies2(alpha+beta)=3(alpha-beta)+4(-2alpha+beta)=0`
`implies2alpha+2beta-3alpha+3beta-8alpha+4beta=0`
`implies-9alpha+9beta=0`
or `alpha=beta`
`implies(alpha)/(beta)=(1)/(1)`
`or alpha:beta=1:1`