Let ` vec a_r=x_r hat i+y_r hat j+z_r hat k ,r=1,2,3` be three mutually perpendicular unit vectors, then the value of `|[x_1,x_2,x_3],[y_1,y_2,y_3],[z_1,z_2,z_3]|` is equal to a. `0` b. `+-1` c. `+-2` d. none of these

If x!=y!=za n d|[x,x^2, 1+x^3],[y ,y^2 ,1+y^3],[z, z^2, 1+z^3]|=0, then the value of x y z is a. 1 b. 2 c. -1 d. 2

If x!=0,y!=0,z!=0 and |[1+x,1,1],[1+y,1+2y,1],[1+z,1+z,1+3z]|=0 , then x^(-1)+y^(-1)+z^(-1) is equal to a.1 b.-1 c.-3 d. none of these

If vec a=x hat i+(x-1) hat j+ hat k and vec b=(x+1) hat i+ hat j+a hat k make an acute angle AAx in R , then find the values of adot

If vec a=x hat i+y hat j+z hat k , vec b=y hat i+z hat j+x hat k and vec c=z hat i+x hat j+y hat k , then vec axx( vec bxx vec c) is

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)

Find the vector equation of a line passing through 3 hat i-5 hat j+7 hat k and perpendicular to the plane 3x-4y+5z=8.

Find the unit vector perpendicular to the plane vec r.(2 hat i+ hat j+2 hat k)=5.

Given a=x//(y-z), b=y//(z-x), and c=z//(x-y),w h e r e .x , y and z are not all zero, then the value of a b+b c+c a is a. 0 b. 1 c. -1"" d. none of these

If |{:(x,x^2,x^3+1),(y,y^2,y^3+1),(z,z^2,z^3+1):}|=a and x ney nez , prove that, xyz=-1.