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Consider three vectors A=hati+hatj-2hatk...

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

A

`1:1`

B

`2:1`

C

`-1:1`

D

`3:1`

Text Solution

Verified by Experts

The correct Answer is:
A
Here `vecA=hati+hatj-2hatk, vecB=hati-hatj+hatk`
`vecC=2hati-3hatj+4hatk`
`vecX=alphavecA+betavecB = alpha(hati+hatj-2hatk)+beta(hati-hatj+hatk)`
`= (alpha+beta) hati + (alpha-beta)hatj+(beta-2alpha) hatk`
As `vecX botvecC` so their dot product is zero .
i.e `vecX . vecC =0`
`((alpha+beta)hati+(alpha-beta)hatj+(beta-2alpha)hatk).(2hati-3hatj+4hatk)=0`
`2alpha+2beta-3alpha+3beta+4beta-8alpha=0`
`-9alpha+9beta=0`
`9alpha=9betaor (alpha)/(beta)=1 or alpha: beta=1:1`
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