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 `0` b. `+-1` c. `+-2` d. none of these

Let overset(to)(a_(r))=x_(r )hat(i)+y_(r )hat(j)+z_(r )hat(k),r=1,2,3 three mutually prependicular 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

Evaluate |^x C_1^x C_2^x C_3^y C_1^y C_2^y C_3^z C_1^z C_2^z C_3|

The value of |(1,1,1),(2x,2y,2z),(3x,3y,3z)| is ____

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

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

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

Let L_1 be the line vec r_1=2 hat i+ hat j- hat k+lambda(i+2 hat k) and let L_2 be the line vec r_2=3 hat i+ hat j+mu(i+ hat j- hat k) . Let pi be the plane which contains the line L_1 and is parallel to L_2dot The distance of the plane pi from the origin is a. sqrt(6) b. 1//7 c. sqrt(2//7) d. none of these

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 ka n d vec c=c_1 hat i+c_2 hat j+c_3 hat k be three nonzero vectors such that vec c is a unit vector perpendicular to both vec aa n d vec bdot If the angle between vec aa n d vec b is pi//6 , then the value of |a_1b_1c_1a_2b_2c_2a_3b_3c_3| is a. 0 b. 1 c. 1/4(a_1 2+a2 2+a3 2)(b1 2+b2 2+b3 2) d. 3/4(a1 2+a2 2+a3 2)(b1 2+b2 2+b3 2)