Show that `|{:(x,y,z),(2x+2a,2y+2b,2z+2c),(a,b,c):}|=0`

Show that {:|(x+2a,y+2b,z+2c),(x,y,z),(a,b,c)|:} =0

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

Show that |{:(a,b,c),(a+2x,b+2y,c+2z),(x,y,z):}|=0

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

Without expanding , show that |(a,b,c),(a+2x,b+2y,c+2z),(x,y,z)|= 0 .

Show that abs((a,b,c),(a+2x,b+2y,c+2z),(x,y,z)) =0

Show that |(0,x,y,z),(-x,0,c,b),(-y,-c,0,a),(-z,-b,-a,0)|=(ax-by+cz)^2

If a x_1^2+b y_1^2+c z_1^2=a x_2 ^2+b y_2 ^2+c z_2 ^2=a x_3 ^2+b y_3 ^2+c z_3 ^2=d ,a x_2 x_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_1, y_1, z_1), (x_2, y_2, z_2), (x_3,y_3,z_3)|=(d-f){((d+2f))/(a b c)}^(1//2)

If a x_1^2+b y_1^2+c z_1^2=a x_2 ^2+b y_2 ^2+c z_2 ^2=a x_3 ^2+b y_3 ^2+c z_3 ^2=d ,a x_2 x_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_1, y_1, z_1), (x_2, y_2, z_2), (x_3,y_3,z_3)|=(d-f){((d+2f))/(a b c)}^(1//2)

Using the properties of determinants prove that |{:(a+b+2c,a,b),(c,b+c+2a,b),(c,a,c+a+2b):}|=2(a+b+c)^3 or |{:(x+y+2z,x,y),(z,y+z+2x,y),(z,x,z+x+2y):}|=2(x+y+z)^3