the area of triangle whose vertices are (1,2,3),(2,5-1) and (-1,1,2) is

Options

A. `(sqrt(155))/(2)` square units
B. `(sqrt(175))/(2)` square units
C. `(sqrt(155))/(4)` square units
D. `(sqrt(175))/(4)` square units

Answer

A

Explanation

Let `vec(O)A=hat(i)+2hat(j)+3hat(k), vec(O)B=2hat(i)+5hat(j)-hat(k)` and `vec(O)C=-hat(i)+hat(j)+2hat(k)` be three position vectors.
Now, `vec(A)B=vec(O)B-vec(O)A=hat(i)+3hat(j)-4hat(k)`
and `vec(A)C=vec(O)C-vec(O)A=-2hat(i)-hat(j)-hat(k)`
`:. ` Area of `Delta=(1)/(2)|vec(A)Bxxvec(A)C|`
Now, `vec(A)Bxxvec(A)C=|(hat(i), hat(j), hat(k)),(1,3,-4),(-2,-1,-1)|`
`=hat(i) (-3-4)-hat(j)(-1-8)+hat(k)(-1+6)`
`=-7hat(i)+9hat(j)+5hat(k)`
Now, `|vec(A)Bxxvec(A)C|=sqrt(155)`
`:.` Required Area `=(sqrt(155))/(2)`