The triad (x ,y ,z ) of real number such that (3i-j+2k )= (2i+3j-k )x+ (i-2j+2k )y+ (-2i+j-2k )z is

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Accepted Answer

Correct answer: C. (c) To solve the given problem, we start by equating the vectors on both sides of the equation: (3i-j+2k )= (2i+3j-k )x+ (i-2j+2k )y+ (-2i+j-2k )z This equation can be split into component equations: 1.3=2x+y-2z 2.-1=3x-2y+z 3.2=-x+2y-2z We will solve this system of equations step-by-step. Step 1: Solve for x from equation (2): -1=3x-2y+z ⇒3x=2y-z-1 ⇒x= 2y-z-13 Step 2: Substitute x into equations (1) and (3): From equation (1): 3=2 ( 2y-z-13 )+y-2z9=4y-2z-2+3y-6z9=7y-8z-2 7y-8z=11 (Equation A) From equation (3): 2=- ( 2y-z-13 )+2y-2z6=-2y+z+1+6y-6z7y-5z=5 ( EquationB ) Step 3: Solve equations (A) and (B) simultaneously: Multiply equation (A) by 5 and equation (B) by 8 to eliminate y : 35y-40z=55 ( EquationC )56y-40z=40 ( EquationD ) Subtract equation (C) from equation (D): (56y-40z )- (35y-40z )=40-5521y=-15 ⇒y=- 1521=- 57 Substitute y=- 57 back into equation (A): 7(- 57)-8z=11-5-8z=11-8z=16 ⇒z=-2 Finally, substitute y=- 57 and z=-2 into the expression for x : x= 2(- 57)- (-2 )-13 x= - 107+2-13x= - 107+ 1473x= 47 Therefore, the solution to the system of equations is (x ,y ,z )=( 47 ,- 57 ,-2). Comparing with the given options, we see that the closest match is option (c): (2 ,5 ,3 )

$The triad (x, y, z) of real number such that (3 i - j + 2 k) = (2 i + 3 j - k) x + (i - 2 j + 2 k) y + (- 2 i + j - 2 k) z is$

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