Find a unit vector perpendicular to the plane `A B C ,` where the coordinates of `A ,B` and C are `A(3,-1,2),B(1,-1,-3)` and `C(4,-3,1)dot`

Find a unit vector perpendicular to the plane ABC, where the coordinates of A ,\ B ,\ a n d\ C are A(3,\ -1,\ 2),\ B(1,\ -1,\ -3)a n d\ C(4,\ -3,\ 1)dot

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

Using vectors, find the area of the triangle whose vertices are: A (3, ̶ 1, 2), B (1, ̶ 1, ̶ 3) and C ( 4, ̶ 3, 1)

Find the area of the quadrilateral A B C D having vertices A(1,1),B(7,-3),C(12 ,2), and D(7, 21)dot

Find a unit vector , C=AxxB is a vector perpendicular to both A and B Henc , a unit vector hatn perpendicular to A and B can be Written as hatn=(C)/(C)=(AxxB)/(|AxxB|) Here, AxxB=|underset(1,-1,1)underset(2,3,1)hati,hatj,hatk| =hati(3+1)+hatj(1-2)+hatk(-2-3) =4hati-hatj-5hatk Further, |AxxB|=sqrt((4)^(2)+(-1)^(2)+(-5)^(2))=sqrt(42) therefore"the desired unit vector is "hatn=(AxxB)/(|AxxB|) or hatn=(1)/(sqrt(42))(4hati -hatj+5hatk)

Let vec a = a_1 hat i + a_2 hat j+ a_3 hat k;vec b = b_1 hat i+ b_2 hat j+ b_3 hat k ; vec c= c_1hat i + c_2 hat j+ c_3 hat k be three non-zero vectors such that vec c is a unit vector perpendicular to both vec a & vec b . If the angle between vec a and vec b is pi/6 , then |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|^2=

A B C D is parallelogram. If the coordinates of A , B , C are (-2,-1),(3,0) and (1,-2) respectively, find the coordinates of Ddot

A B C D is parallelogram. If the coordinates of A , B , C are (-2,-1),(3,0) and (1,-2) respectively, find the coordinates of Ddot

Let vec a=a_1 hat i+a_2 hat j+a_3 hat k , vec b=b_1 hat i+b_2 hat j+b_3 hat k and vec c=c_1 hat i+c_2 hat j+c_3 hat k be three non-zero vectors such that vec c is a unit vector perpendicular to both vec a and vec b . If the angle between a and b is pi/6, then prove that |[a_1,a_2,a_3],[b_1,b_2,b_3],[c_1,c_2,c_3]|^2=1/4(a_1 ^2+a_2 ^2+a_3 ^2)(b_1 ^2+b_2 ^2+b_3 ^2)

Let vec a=a_1 hat i+a_2 hat j+a_2 hat k , vec b=b_1 hat i+a_2 hat j+b_2 hat k , and vec c=c_1 hat i+c_2 hat j+c_2 hat k , be three non-zero vectors such that vec c is a unit vector perpendicular to both vectors vec a and vec b . If the angle between a and b is pi//6, then |[a_1,a_2,a_3],[b_1,b_2,b_3],[c_1,c_2,c_3]|^2 is equal to