If `a x1 2+b y1 2+c z1 2=a x2 2+b y2 2+c z2 2=a x3 2+b y3 2+c z3 2=d ,a x2 3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1=a x_1x_2+b y_1y_2+c z_1z_2=f,` then prove that `|x_1y_1z_1x_2y_2z_2x_3y_3z_3|=(d-f){((d+2f))/(a b c)}^(1//2)`

Prove that |(x+2a,y+2b,z+2c),(x,y,z),(a,b,c)|=0

it x_(1)^(2) +2y_(1)^(2)+3z_(1)^(2)=x_(2)^(2)+2y_(2)^(2)+3z_(2)^(2)=x_(3)^(2)+2y_(3)^(2)+3z_(3)^(2)=2 " and " x_(2)x_(3) +2y_(2)y_(3)+3z_(2)z_(3)=x_(3)x_(1)+2y_(3)y_(1)+3z_(3)z_(1)=x_(1)x_(2)+2y_(1)y_(2)+3z_(1)z_(2)=1 Then find the value of |{:(x_(1),,y_(1),,z_(1)),(x_(2),,y_(2),,z_(2)),(x_(3),,y_(3),,z_(3)):}|

Prove that |(1,x,x^2),(1,y,y^2),(1,z,z^2)|=(x-y)(y-z)(z-x)

The system of linear equations x + y + z = 2 2x + 3y + 2z = 5 2x + 3y + (a^(2) - 1)z = a + 1

Prove that |a x-b y-c z a y+b x c x+a z a y+b x b y-c z-a x b z+c y c x+a z b z+c y c z-a x-b y|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot

Show that |(x,y,z),(x^(2),y^(2),z^(2)),(x^(3),y^(3),z^(3))|=xyz (x-y) (y-z) (z-x)

Sow that x_1 hat i+y_1 hat j+z_1 hat k ,x_2 hat i+y_2 hat j+z_2 hat k ,a n dx_3 hat i+y_3 hat j+z_3 hat k , are non-coplanar if |x_1|>|y_1|+|z_1|,|y_2|>|x_2|+|z_2|a n d|z_3|>|x_3|+|y_3| .

If x=(log)_(2a)a , y=(log)_(3a)2a ,z=(log)_(4a)3a ,prove that 1+x y z=2y z

Show that |(1,1,1),(x,y,z),(x^(2),y^(2),z^(2))|=(x-y)(y-z)(z-x)

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|