Home
Class 12
MATHS
Show that abs(((a-x)^2,(a-y)^2,(a-z)^2...

Show that
`abs(((a-x)^2,(a-y)^2,(a-z)^2) , ((b-x)^2,(b-y)^2,(b-z)^2) , ((c-x)^2,(c-y)^2, (c-z)^2)) = 2(a-b)(b-c)(c-a)(x-y)(y-z)(z-x)`

Text Solution

Verified by Experts

` D = D_(1) D_(2)` (see determinants)
`=2|{:(a^(2),a,1),(b^(2),b,1),(c^(2),c,1):}||{:(1,x,x^(2)),(1,y,y^(2)),(1,z,z^(2)):}| = 0`
since `vecA, vecB and vecC` are non- coplanar, `D_(1) ne 0`,
`D_(2)= 0 or |{:(x^(2),x,1),(y^(2),y,1),(z^(2),z,1):}|=0`
or `vecX , vecY and vecZ` are coplanar.
Promotional Banner

Topper's Solved these Questions

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE PUBLICATION|Exercise Exercises MCQ|134 Videos

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE PUBLICATION|Exercise Reasoning type|8 Videos

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE PUBLICATION|Exercise Exercise 2.3|18 Videos

  • DETERMINANTS

    CENGAGE PUBLICATION|Exercise All Questions|262 Videos

  • DIFFERENTIAL EQUATIONS

    CENGAGE PUBLICATION|Exercise All Questions|578 Videos

Similar Questions

Explore conceptually related problems

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

Prove that |{:((b+x)(c+x),,(c+x)(a+x),,(a+x)(b+x)),((b+y)(c+y) ,,(c+y)(a+y),,(a+y)(b+y)),((b+z)(c+z),, (c+z)(a+z),,(a+z)(b+x)):}| = (b-c) (c-a) (a-b) (y-z) (z-x) (x-y)

Prove that the value of each the following determinants is zero: |[(a^x+a^(-x))^2,(a^x-a^(-x))^2 ,1],[(b^y+b^(-y))^2,(b^y-b^(-y))^2 ,1],[(c^z+c^(-z))^2,(c^z-c^(-z))^2, 1]|

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

prove that, |{:(x,b,c),(a,y,c),(a,b,z):}|=(x-a)(y-b)(z-c)((x)/(x-a)+(y)/(y-b)+(z)/(z-c)-2)

Show that a^(log_(a^2)^(x))xxb^(log_(b^2)^(y))xxc^(log_(c^2)^(z))=sqrt(xyz)

If x/a = y/b = z/c , then show that (x^(2)-yz)/(a^(2)-bc) = (y^(2)-zx)/(b^(2) - ca) = (z^(2)-xy)/(c^(2)-ab)

(a) If x + y + z=0, show that x ^(3) + y ^(3) + z ^(3)= 3 xyz. (b) Show that (a-b)^(3) + (b-c) ^(3) + (c-a)^(3) =3 (a-b) (b-c) (c-a)

If a/(y+z) = b/(z + x) = c/(x+y) , then prove that (a(b-c))/(y^(2)-z^(2)) = (b(c-a))/(z^(2)-x^(2)) = (c(a-b))/(x^(2)-y^(2)) .

By using properties of determinants , show that : (i) {:|( a-b-c, 2a, 2a),( 2b, b-c-a, 2b ) , ( 2c,2c, c-a-b) |:} = ( a+b+c)^(3) (ii) {:|( x+y+2z, x,y ),( z, y+z+2x, y ),( z,x,z+x+2y )|:} =2( x+y+z)^(3)