The values of such that (x, y, z) (0,0,0) and ( i + j +3 k ) x+(3 i -3 j + k ) y+(-4 i +5 j ) z= ( i x+ j y+ k z ) are

Step-by-step solution

Given, aligned &( i + j +3 k ) x+(3 i -3 j + k ) y &+(-4 i +5 j ) z= ( i x+ j y+ k z) aligned On equating the coefficients of i , j and k both sides, we have aligned & x+3 y-4 z= x & x-3 y+5 z= y aligned and 3 x+y+0= z Above three equations can be rewritten as array r (1- ) x+3 y-4 z=0 x-(3+ ) y+5 z=0 3 x+y- z=0 array This is homogeneous system of equations in three variables x, y and z . It is consistent and have non-zero solution. i.e. (x, y, z) (0,0,0) , if determinant of coefficient matrix is zero. | array ccc