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The equation of the plane passing throug...

The equation of the plane passing through three points A,B,C with position vectors `-6hat(i)+3hat(j)+2hat(k),3hat(i)-2hat(j)+4hat(k),5hat(i)+7hat(j)+3hat(k)`, is

A

`r*(hat(i)-hat(j)-7hat(k))+23=0`

B

`r*(hat(i)+hat(j)+7hat(k))=23`

C

`r*(hat(i)+hat(j)-7hat(k))+23=0`

D

`r*(hat(i)-hat(j)-7hat(k))=23`

Text Solution

Verified by Experts

The correct Answer is:
A
Equation of the plane passing through three points A, B, C with position vectors a,b,c is
`r*(axxb+bxxc+cxxa)=a*(bxxc)`
Let a,b,c represents the position vectors of given points
Then, `axxb+bxxc+cxxa`
`=|{:(hat(i),hat(j),hat(k)),(-6,3,2),(3,-2,4):}|+|{:(hat(i),hat(j),hat(k)),(3,-2,4),(5,7,3):}|+|{:(hat(i),hat(j),hat(k)),(5,7,3),(-6,3,2):}|`
`{hat(i)(12+4)-hat(j)(-24-6)+hat(k)(12-9)}`
`+[hat(i)(-6-28)-hat(j)(9-20)+hat(k)(21+10)]`
`+{hat(i)(14-9)-hat(j)(10+18)+hat(k)(15+42)}`
`=16hat(i)+30hat(j)+3hat(k)-34hat(i)+11hat(j)+3hat(k)+5hat(i)-28hat(j)+57hat(k)=-13hat(i)+13hat(j)+91hat(k)`
`anda*(bxxc)=|{:(-6,3,2),(3,-2,4),(5,7,3):}|`
`=-6(-34)-3(-11)+2(31)=299`
So, the required equation of the plane is
`r*(-13hat(i)+13hat(j)+91hat(k))=299`
`or" "r*(hat(i)-hat(j)-7hat(k))=-23`
`or" "r*(hat(i)-hat(j)-7hat(k))+23=0`
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