If x 1 2 3 y 2 1 2 and z 2 3 0 determine the vector x y

If x = (1, 2, -3), y = (2, 1, -2) and z = (-2, 3, 0) determine the vector x + y - 2z.

Given that

x = ( 1 , 2 , -3 ) , y = ( 2 , 1 , -2 ) , z = ( -2 , 3 , 0 )

The vector x + y - 2z = ( 1 , 2 , -3 ) + ( 2 , 1 , -2 ) - 2(-2 , 3 , 0 )

= ( 1 , 2 , -3 ) + ( 2 , 1 , -2 ) - ( -4 , 6 , 0 ) [since ,applying scalar multiplication is 2(a ,b)=2a+2b]

= ( 1+2 , 2+1 , -3+(-2) ) - ( -4 , 6 , 0 ) [ since , ( a, b,c) + ( d , e , f ) = ( a + d , b + e, c+f ) ]

= ( 3 , 3 , -3 - 2 ) - ( -4 , 6 , 0 )

= ( 3 , 3 , -5 ) - ( -4 , 6 , 0 )

= ( 3 - (-4) , 3 - 6 , -5 - 0 ) [ since , ( a, b,c) - ( d , e , f ) = ( a - d , b - e, c - f ) ]

= ( 3 + 4 , -3 , -5 )

= ( 7 , -3 , -5 )

Therefore ,

Vector x + y - 2z = ( 7 , -3 , -5 )