Show that `|1+a1 1 1 1+b1 1 1 1+c|=a b c(1+1/a+1/b+1/c)=a b c+b c+c a+a b`

Prove that |{:(a+1,1,1),(1,b+1,1),(1,1,c+1):}|=abc(1/a+1/b+1/c+1)

Show that two lines a_(1) x + b_(14) y + c_(1) = 0 and a_(2) x + b_(2) y + c_(2) = 0 , where b_(1) , b_(2) ne 0 are (i) Parallel if (a_1)/( b_1) = (a_2)/( b_2) and

Show that two lines a_(1) x + b_(14) y + c_(1) = 0 and a_(2) x + b_(2) y + c_(2) = 0 , where b_(1) , b_(2) ne 0 are Perpendicular if a_(1) a_(2) + b_(1) b_(2) =0 .

Let vec(a)=hati+hatj+hatk,vec(b)=hati and vec( c )=c_(1)hati+c_(2)hatj+c_(3)hatk . Then If c_(2)=-1 and c_(3)=1 . Show that no value of c_(1) can make vec(a),vec(b) and vec( c ) coplanar.

If a x1 2+b y1 2+c z1 2=a x1 2+b y2 2+c z2 2=a x3 2+b y3 2+c z3 2=d\ a n d\ a x_2x_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_1x_3y_3z_3|=(d-f)[(d+2f)/(a b c)]^(//2)(a , b ,\ c!=0)

If (a, 1, 1), (1, b, 1) and (1, 1, c) are coplanar then prove that (1)/(1-a)+(1)/(1-b)+(1)/(1-c)=1 .

Without expanding the determinant, prove that {:[( a, a ^(2), bc ),( b ,b ^(2) , ca),( c, c ^(2) , ab ) ]:} ={:[( 1, a^(2) , a^(3) ),( 1,b^(2) , b^(3) ),( 1, c^(2),c^(3)) ]:}

If the lines ax + y + 1 = 0, x + by + 1 = 0 and x + y + c = 0 are concurrent then prove that, (1)/( 1-a) + (1) /( 1-b) + (1) /( 1-c) = .

If 1/a+1/b+1/c = 1/(a+b+c) then (a+b)(b+c)(c+a)=....... a. a b c b. 1/(a b c) c. 0 d. 1