Find the area of the triangle whose vertices are `A(3,-1,2),\ B(1,-1,-3)a n d\ C(4,-3,1)dot`

Options

A. `(sqrt(165))/(2)`
B. `(sqrt(135))/(2)`
C. 4
D. 2

Answer

A

Explanation

Let the vertices of the `DeltaABC ` are A (3, -1, 2), B (1, -1, -3) and C(4, -3, 1).
Let `vec(OA)=3hat(i)-hat(j)+2hat(k)`,
`vec(OB)=hat(i)-hat(j)-3hat(k) and`
`vec(OC)=4hat(i)-3hat(j)+k`
Area of `DeltaABC=(1)/(2)|vec(AB)xxvec(AC)|`
Now, `vec(AB)=bar(OA)-bar(OB)=2hat(i)+5hat(k)`
`vec(AC)=bar(OA)-bar(OC)=-hat(i)+2hat(j)+hat(k)`
`:.` Required Area `=(1)/(2)||(hat(i), hat(j), hat(k)), (2 , 0, 5),(-1, 2, 1)||`
`=(1)/(2)|hat(i)(-10)-hat(j)(2+5)+hat(k)(4)|`
`=(1)/(2)|-10hat(i)-7hat(j)+4hat(k)|`
`=(1)/(2) sqrt(100+49+16)=(1)/(2)sqrt(165)`sq unit