Find the vector equation of the plane passing thrugh the points (2,5,-3),(-2,-3,5),(5,3,-3).

Options

A. `[r-(2hat(i)+5hat(j)-3hat(k))]*(-4hat(i)-8hat(j)+8hat(k))=0`
B. `[r-(2hat(i)+5hat(j)-3hat(k))]*(3hat(i)-2hat(j))=0`
C. `[r-(2hat(i)+5hat(j)-3hat(k))]*(16hat(i)+24hat(j)+32hat(k))=0`
D. None of the above

Answer

C

Explanation

Let `a=2hat(i)+5hat(j)-3hat(k),b=-2hat(i)-3hat(j)+5hat(k),c=5hat(i)+3hat(j)-3hat(k)`
Then, the vector equation of the plane passing through a, b and c and is given by
`(r-a)*[(b-a)xx(c-a)]=0`
`rArr[r-(2hat(i)+5hat(j)-3hat(k))]*[(-4hat(i)-8hat(j)+8hat(k))xx(3hat(i)-2hat(j))]=0` . . . (i)
Now, `(-4hat(i)-8hat(j)+8hat(k))xx(3hat(i)-2hat(j))=|{:(hat(i),hat(j),hat(k)),(-4,-8,8),(3,-2,0):}|`
`=hat(i)(0+16)-hat(j)(0-24)+hat(k)(8+24)=16hat(i)+24hat(j)+32hat(k)`
Hence, required equation of plane is
`[r-(2hat(i)+5hat(j)-3hat(k))]*(16hat(i)+24hat(j)+32hat(k))=0`